Theorem lt2msq 5839

Description: The square function on nonnegative reals is strictly monotonic.

Hypotheses

Ref	Expression
ltrec.1	|- A e. RR
ltrec.2	|- B e. RR

Assertion

Ref	Expression
lt2msq	|- ((0 <_ A /\ 0 <_ B) -> (A < B <-> (A x. A) < (B x. B)))

Proof of Theorem lt2msq

Step	Hyp	Ref	Expression
1		ltrec.1	. . . . . . . 8 |- A e. RR
2		ltrec.2	. . . . . . . 8 |- B e. RR
3	1, 2, 1	ltmul2 5800	. . . . . . 7 |- (0 < A -> (A < B <-> (A x. A) < (A x. B)))
4	1, 2, 2	ltmul1 5788	. . . . . . 7 |- (0 < B -> (A < B <-> (A x. B) < (B x. B)))
5	3, 4	bi2anan9 631	. . . . . 6 |- ((0 < A /\ 0 < B) -> ((A < B /\ A < B) <-> ((A x. A) < (A x. B) /\ (A x. B) < (B x. B))))
6		anidm 432	. . . . . 6 |- ((A < B /\ A < B) <-> A < B)
7	5, 6	syl5bbr 533	. . . . 5 |- ((0 < A /\ 0 < B) -> (A < B <-> ((A x. A) < (A x. B) /\ (A x. B) < (B x. B))))
8	1, 1	remulcl 5318	. . . . . 6 |- (A x. A) e. RR
9	1, 2	remulcl 5318	. . . . . 6 |- (A x. B) e. RR
10	2, 2	remulcl 5318	. . . . . 6 |- (B x. B) e. RR
11	8, 9, 10	lttr 5569	. . . . 5 |- (((A x. A) < (A x. B) /\ (A x. B) < (B x. B)) -> (A x. A) < (B x. B))
12	7, 11	syl6bi 214	. . . 4 |- ((0 < A /\ 0 < B) -> (A < B -> (A x. A) < (B x. B)))
13	2, 1, 2	lemul2 5802	. . . . . . . . 9 |- (0 < B -> (B <_ A <-> (B x. B) <_ (B x. A)))
14	2, 1, 1	lemul1 5801	. . . . . . . . 9 |- (0 < A -> (B <_ A <-> (B x. A) <_ (A x. A)))
15	13, 14	bi2anan9r 632	. . . . . . . 8 |- ((0 < A /\ 0 < B) -> ((B <_ A /\ B <_ A) <-> ((B x. B) <_ (B x. A) /\ (B x. A) <_ (A x. A))))
16		anidm 432	. . . . . . . 8 |- ((B <_ A /\ B <_ A) <-> B <_ A)
17	15, 16	syl5bbr 533	. . . . . . 7 |- ((0 < A /\ 0 < B) -> (B <_ A <-> ((B x. B) <_ (B x. A) /\ (B x. A) <_ (A x. A))))
18	2, 1	remulcl 5318	. . . . . . . 8 |- (B x. A) e. RR
19	10, 18, 8	letr 5572	. . . . . . 7 |- (((B x. B) <_ (B x. A) /\ (B x. A) <_ (A x. A)) -> (B x. B) <_ (A x. A))
20	17, 19	syl6bi 214	. . . . . 6 |- ((0 < A /\ 0 < B) -> (B <_ A -> (B x. B) <_ (A x. A)))
21	2, 1	lenlt 5561	. . . . . 6 |- (B <_ A <-> -. A < B)
22	10, 8	lenlt 5561	. . . . . 6 |- ((B x. B) <_ (A x. A) <-> -. (A x. A) < (B x. B))
23	20, 21, 22	3imtr3g 551	. . . . 5 |- ((0 < A /\ 0 < B) -> (-. A < B -> -. (A x. A) < (B x. B)))
24	23	a3d 75	. . . 4 |- ((0 < A /\ 0 < B) -> ((A x. A) < (B x. B) -> A < B))
25	12, 24	impbid 515	. . 3 |- ((0 < A /\ 0 < B) -> (A < B <-> (A x. A) < (B x. B)))
26		breq1 2618	. . . . 5 |- (0 = A -> (0 < B <-> A < B))
27	26	adantr 389	. . . 4 |- ((0 = A /\ 0 < B) -> (0 < B <-> A < B))
28		0re 5423	. . . . . 6 |- 0 e. RR
29	28, 2, 2	ltmul2 5800	. . . . 5 |- (0 < B -> (0 < B <-> (B x. 0) < (B x. B)))
30		opreq2 3964	. . . . . . 7 |- (0 = A -> (A x. 0) = (A x. A))
31	30	breq1d 2625	. . . . . 6 |- (0 = A -> ((A x. 0) < (B x. B) <-> (A x. A) < (B x. B)))
32	2	recn 5297	. . . . . . . . 9 |- B e. CC
33	32	mul01 5414	. . . . . . . 8 |- (B x. 0) = 0
34	1	recn 5297	. . . . . . . . 9 |- A e. CC
35	34	mul01 5414	. . . . . . . 8 |- (A x. 0) = 0
36	33, 35	eqtr4 1496	. . . . . . 7 |- (B x. 0) = (A x. 0)
37	36	breq1i 2622	. . . . . 6 |- ((B x. 0) < (B x. B) <-> (A x. 0) < (B x. B))
38	31, 37	syl5bb 531	. . . . 5 |- (0 = A -> ((B x. 0) < (B x. B) <-> (A x. A) < (B x. B)))
39	29, 38	sylan9bbr 540	. . . 4 |- ((0 = A /\ 0 < B) -> (0 < B <-> (A x. A) < (B x. B)))
40	27, 39	bitr3d 529	. . 3 |- ((0 = A /\ 0 < B) -> (A < B <-> (A x. A) < (B x. B)))
41		breq1 2618	. . . . . . 7 |- (0 = B -> (0 <_ A <-> B <_ A))
42	41	adantl 388	. . . . . 6 |- ((0 < A /\ 0 = B) -> (0 <_ A <-> B <_ A))
43	28, 1, 1	lemul2 5802	. . . . . . 7 |- (0 < A -> (0 <_ A <-> (A x. 0) <_ (A x. A)))
44		opreq2 3964	. . . . . . . . 9 |- (0 = B -> (B x. 0) = (B x. B))
45	44	breq1d 2625	. . . . . . . 8 |- (0 = B -> ((B x. 0) <_ (A x. A) <-> (B x. B) <_ (A x. A)))
46	35, 33	eqtr4 1496	. . . . . . . . 9 |- (A x. 0) = (B x. 0)
47	46	breq1i 2622	. . . . . . . 8 |- ((A x. 0) <_ (A x. A) <-> (B x. 0) <_ (A x. A))
48	45, 47	syl5bb 531	. . . . . . 7 |- (0 = B -> ((A x. 0) <_ (A x. A) <-> (B x. B) <_ (A x. A)))
49	43, 48	sylan9bb 539	. . . . . 6 |- ((0 < A /\ 0 = B) -> (0 <_ A <-> (B x. B) <_ (A x. A)))
50	42, 49	bitr3d 529	. . . . 5 |- ((0 < A /\ 0 = B) -> (B <_ A <-> (B x. B) <_ (A x. A)))
51	50, 21, 22	3bitr3g 553	. . . 4 |- ((0 < A /\ 0 = B) -> (-. A < B <-> -. (A x. A) < (B x. B)))
52	51	con4bid 523	. . 3 |- ((0 < A /\ 0 = B) -> (A < B <-> (A x. A) < (B x. B)))
53		pm5.21 676	. . . 4 |- ((-. A < B /\ -. (A x. A) < (B x. B)) -> (A < B <-> (A x. A) < (B x. B)))
54	2	ltnr 5593	. . . . 5 |- -. B < B
55		breq1 2618	. . . . . . 7 |- (0 = B -> (0 < B <-> B < B))
56	55	bicomd 520	. . . . . 6 |- (0 = B -> (B < B <-> 0 < B))
57	56, 26	sylan9bbr 540	. . . . 5 |- ((0 = A /\ 0 = B) -> (B < B <-> A < B))
58	54, 57	mtbii 715	. . . 4 |- ((0 = A /\ 0 = B) -> -. A < B)
59	10	ltnr 5593	. . . . 5 |- -. (B x. B) < (B x. B)
60	44	breq1d 2625	. . . . . . 7 |- (0 = B -> ((B x. 0) < (B x. B) <-> (B x. B) < (B x. B)))
61	60	bicomd 520	. . . . . 6 |- (0 = B -> ((B x. B) < (B x. B) <-> (B x. 0) < (B x. B)))
62	61, 38	sylan9bbr 540	. . . . 5 |- ((0 = A /\ 0 = B) -> ((B x. B) < (B x. B) <-> (A x. A) < (B x. B)))
63	59, 62	mtbii 715	. . . 4 |- ((0 = A /\ 0 = B) -> -. (A x. A) < (B x. B))
64	53, 58, 63	sylanc 471	. . 3 |- ((0 = A /\ 0 = B) -> (A < B <-> (A x. A) < (B x. B)))
65	25, 40, 52, 64	ccase 754	. 2 |- (((0 < A \/ 0 = A) /\ (0 < B \/ 0 = B)) -> (A < B <-> (A x. A) < (B x. B)))
66	28, 1	leloe 5558	. 2 |- (0 <_ A <-> (0 < A \/ 0 = A))
67	28, 2	leloe 5558	. 2 |- (0 <_ B <-> (0 < B \/ 0 = B))
68	65, 66, 67	syl2anb 455	1 |- ((0 <_ A /\ 0 <_ B) -> (A < B <-> (A x. A) < (B x. B)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 955   e. wcel 957   class class class wbr 2615  (class class class)co 3958  RRcr 5216  0cc0 5217   x. cmul 5222   <_ cle 5278   < clt 5469

This theorem is referenced by:  le2msq 5840  msq11 5841  lt2msqt 5844  lt2sq 6569  sqrlem6 6623  sqrlem12 6629

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-inf2 4608

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-nel 1586  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-pss 2052  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-om 3128  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194  df-rdg 3927  df-opr 3960  df-oprab 3961  df-1st 4072  df-2nd 4073  df-1o 4126  df-oadd 4128  df-omul 4129  df-er 4254  df-ec 4256  df-qs 4259  df-en 4360  df-dom 4361  df-sdom 4362  df-ni 4983  df-pli 4984  df-mi 4985  df-lti 4986  df-plpq 5018  df-mpq 5019  df-enq 5020  df-nq 5021  df-plq 5022  df-mq 5023  df-rq 5024  df-ltq 5025  df-1q 5026  df-np 5069  df-1p 5070  df-plp 5071  df-mp 5072  df-ltp 5073  df-plpr 5147  df-mpr 5148  df-enr 5149  df-nr 5150  df-plr 5151  df-mr 5152  df-ltr 5153  df-0r 5154  df-1r 5155  df-m1r 5156  df-c 5223  df-0 5224  df-1 5225  df-i 5226  df-r 5227  df-plus 5228  df-mul 5229  df-lt 5230  df-sub 5339  df-neg 5341  df-pnf 5470  df-mnf 5471  df-xr 5472  df-ltxr 5473  df-le 5474

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