Theorem sqrlem12 6698

Description: Lemma for square root theorem.

Hypotheses

Ref	Expression
sqrlem1.1	⊢ A ∈ ℝ
sqrlem1.2	⊢ 0 < A
sqrlem9.3	⊢ B ∈ ℝ
sqrlem9.4	⊢ C ∈ ℝ
sqrlem9.5	⊢ 0 < B
sqrlem9.6	⊢ A < (B · B)
sqrlem9.7	⊢ C = ((B + (A / B)) / (1 + 1))
sqrlem12.8	⊢ S = {x ∈ ℝ∣(0 ≤ x ⋀ (x · x) ≤ A)}

Assertion

Ref	Expression
sqrlem12	⊢ (D ∈ S → D < C)

Distinct variable groups:   x,A   x,B   x,S   x,C   x,D

Proof of Theorem sqrlem12

Step	Hyp	Ref	Expression
1		sqrlem1.1	. . . . . 6 ⊢ A ∈ ℝ
2		sqrlem1.2	. . . . . 6 ⊢ 0 < A
3		sqrlem12.8	. . . . . 6 ⊢ S = {x ∈ ℝ∣(0 ≤ x ⋀ (x · x) ≤ A)}
4	1, 2, 3	sqrlem4 6690	. . . . 5 ⊢ (D ∈ S ↔ (D ∈ ℝ ⋀ (0 ≤ D ⋀ (D · D) ≤ A)))
5	4	pm3.27bi 326	. . . 4 ⊢ (D ∈ S → (0 ≤ D ⋀ (D · D) ≤ A))
6	5	pm3.26d 321	. . 3 ⊢ (D ∈ S → 0 ≤ D)
7	4	pm3.26bi 322	. . . 4 ⊢ (D ∈ S → D ∈ ℝ)
8		0re 5453	. . . . 5 ⊢ 0 ∈ ℝ
9		leloet 5531	. . . . 5 ⊢ ((0 ∈ ℝ ⋀ D ∈ ℝ) → (0 ≤ D ↔ (0 < D ⋁ 0 = D)))
10	8, 9	mpan 699	. . . 4 ⊢ (D ∈ ℝ → (0 ≤ D ↔ (0 < D ⋁ 0 = D)))
11	7, 10	syl 10	. . 3 ⊢ (D ∈ S → (0 ≤ D ↔ (0 < D ⋁ 0 = D)))
12	6, 11	mpbid 195	. 2 ⊢ (D ∈ S → (0 < D ⋁ 0 = D))
13	5	pm3.27d 325	. . . . . . 7 ⊢ (D ∈ S → (D · D) ≤ A)
14		sqrlem9.3	. . . . . . . . 9 ⊢ B ∈ ℝ
15		sqrlem9.4	. . . . . . . . 9 ⊢ C ∈ ℝ
16		sqrlem9.5	. . . . . . . . 9 ⊢ 0 < B
17		sqrlem9.6	. . . . . . . . 9 ⊢ A < (B · B)
18		sqrlem9.7	. . . . . . . . 9 ⊢ C = ((B + (A / B)) / (1 + 1))
19	1, 2, 14, 15, 16, 17, 18	sqrlem11 6697	. . . . . . . 8 ⊢ A < (C · C)
20		axmulrcl 5287	. . . . . . . . . 10 ⊢ ((D ∈ ℝ ⋀ D ∈ ℝ) → (D · D) ∈ ℝ)
21	20	anidms 437	. . . . . . . . 9 ⊢ (D ∈ ℝ → (D · D) ∈ ℝ)
22	15, 15	remulcl 5348	. . . . . . . . . 10 ⊢ (C · C) ∈ ℝ
23		lelttrt 5536	. . . . . . . . . 10 ⊢ (((D · D) ∈ ℝ ⋀ A ∈ ℝ ⋀ (C · C) ∈ ℝ) → (((D · D) ≤ A ⋀ A < (C · C)) → (D · D) < (C · C)))
24	1, 22, 23	mp3an23 912	. . . . . . . . 9 ⊢ ((D · D) ∈ ℝ → (((D · D) ≤ A ⋀ A < (C · C)) → (D · D) < (C · C)))
25	7, 21, 24	3syl 20	. . . . . . . 8 ⊢ (D ∈ S → (((D · D) ≤ A ⋀ A < (C · C)) → (D · D) < (C · C)))
26	19, 25	mpan2i 703	. . . . . . 7 ⊢ (D ∈ S → ((D · D) ≤ A → (D · D) < (C · C)))
27	13, 26	mpd 26	. . . . . 6 ⊢ (D ∈ S → (D · D) < (C · C))
28	27	adantr 391	. . . . 5 ⊢ ((D ∈ S ⋀ 0 < D) → (D · D) < (C · C))
29	1, 2, 14, 15, 16, 17, 18	sqrlem9 6695	. . . . . . 7 ⊢ 0 < C
30		breq2 2636	. . . . . . . . . . 11 ⊢ (D = if(D ∈ ℝ, D, 0) → (0 < D ↔ 0 < if(D ∈ ℝ, D, 0)))
31	30	anbi1d 620	. . . . . . . . . 10 ⊢ (D = if(D ∈ ℝ, D, 0) → ((0 < D ⋀ 0 < C) ↔ (0 < if(D ∈ ℝ, D, 0) ⋀ 0 < C)))
32		breq1 2635	. . . . . . . . . . 11 ⊢ (D = if(D ∈ ℝ, D, 0) → (D < C ↔ if(D ∈ ℝ, D, 0) < C))
33		opreq12 3984	. . . . . . . . . . . . 13 ⊢ ((D = if(D ∈ ℝ, D, 0) ⋀ D = if(D ∈ ℝ, D, 0)) → (D · D) = ( if(D ∈ ℝ, D, 0) · if(D ∈ ℝ, D, 0)))
34	33	anidms 437	. . . . . . . . . . . 12 ⊢ (D = if(D ∈ ℝ, D, 0) → (D · D) = ( if(D ∈ ℝ, D, 0) · if(D ∈ ℝ, D, 0)))
35	34	breq1d 2642	. . . . . . . . . . 11 ⊢ (D = if(D ∈ ℝ, D, 0) → ((D · D) < (C · C) ↔ ( if(D ∈ ℝ, D, 0) · if(D ∈ ℝ, D, 0)) < (C · C)))
36	32, 35	bibi12d 632	. . . . . . . . . 10 ⊢ (D = if(D ∈ ℝ, D, 0) → ((D < C ↔ (D · D) < (C · C)) ↔ ( if(D ∈ ℝ, D, 0) < C ↔ ( if(D ∈ ℝ, D, 0) · if(D ∈ ℝ, D, 0)) < (C · C))))
37	31, 36	imbi12d 629	. . . . . . . . 9 ⊢ (D = if(D ∈ ℝ, D, 0) → (((0 < D ⋀ 0 < C) → (D < C ↔ (D · D) < (C · C))) ↔ ((0 < if(D ∈ ℝ, D, 0) ⋀ 0 < C) → ( if(D ∈ ℝ, D, 0) < C ↔ ( if(D ∈ ℝ, D, 0) · if(D ∈ ℝ, D, 0)) < (C · C)))))
38	8	elimel 2404	. . . . . . . . . . 11 ⊢ if(D ∈ ℝ, D, 0) ∈ ℝ
39	38, 15	lt2msq 5887	. . . . . . . . . 10 ⊢ ((0 ≤ if(D ∈ ℝ, D, 0) ⋀ 0 ≤ C) → ( if(D ∈ ℝ, D, 0) < C ↔ ( if(D ∈ ℝ, D, 0) · if(D ∈ ℝ, D, 0)) < (C · C)))
40	8, 38	ltle 5593	. . . . . . . . . 10 ⊢ (0 < if(D ∈ ℝ, D, 0) → 0 ≤ if(D ∈ ℝ, D, 0))
41	8, 15	ltle 5593	. . . . . . . . . 10 ⊢ (0 < C → 0 ≤ C)
42	39, 40, 41	syl2an 457	. . . . . . . . 9 ⊢ ((0 < if(D ∈ ℝ, D, 0) ⋀ 0 < C) → ( if(D ∈ ℝ, D, 0) < C ↔ ( if(D ∈ ℝ, D, 0) · if(D ∈ ℝ, D, 0)) < (C · C)))
43	37, 42	dedth 2393	. . . . . . . 8 ⊢ (D ∈ ℝ → ((0 < D ⋀ 0 < C) → (D < C ↔ (D · D) < (C · C))))
44	7, 43	syl 10	. . . . . . 7 ⊢ (D ∈ S → ((0 < D ⋀ 0 < C) → (D < C ↔ (D · D) < (C · C))))
45	29, 44	mpan2i 703	. . . . . 6 ⊢ (D ∈ S → (0 < D → (D < C ↔ (D · D) < (C · C))))
46	45	imp 350	. . . . 5 ⊢ ((D ∈ S ⋀ 0 < D) → (D < C ↔ (D · D) < (C · C)))
47	28, 46	mpbird 196	. . . 4 ⊢ ((D ∈ S ⋀ 0 < D) → D < C)
48	47	ex 373	. . 3 ⊢ (D ∈ S → (0 < D → D < C))
49		breq1 2635	. . . . 5 ⊢ (0 = D → (0 < C ↔ D < C))
50	29, 49	mpbii 193	. . . 4 ⊢ (0 = D → D < C)
51	50	a1i 8	. . 3 ⊢ (D ∈ S → (0 = D → D < C))
52	48, 51	jaod 426	. 2 ⊢ (D ∈ S → ((0 < D ⋁ 0 = D) → D < C))
53	12, 52	mpd 26	1 ⊢ (D ∈ S → D < C)

Syntax hints:   → wi 3   ↔ wb 146   ⋁ wo 222   ⋀ wa 223   = wceq 960   ∈ wcel 962  {crab 1655   ifcif 2371   class class class wbr 2632  (class class class)co 3977  ℝcr 5246  0cc0 5247  1c1 5248   + caddc 5250   · cmul 5252   / cdiv 5307   ≤ cle 5308   < clt 5499

This theorem is referenced by:  sqrlem13 6699

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-9 969  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1129  ax-10o 1146  ax-16 1216  ax-11o 1224  ax-ext 1466  ax-rep 2706  ax-sep 2716  ax-nul 2723  ax-pow 2756  ax-pr 2793  ax-un 2880  ax-inf2 4637

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 780  df-3an 781  df-ex 985  df-sb 1178  df-eu 1388  df-mo 1389  df-clab 1471  df-cleq 1476  df-clel 1479  df-ne 1594  df-nel 1595  df-ral 1656  df-rex 1657  df-reu 1658  df-rab 1659  df-v 1819  df-sbc 1949  df-csb 2010  df-dif 2058  df-un 2059  df-in 2060  df-ss 2062  df-pss 2064  df-nul 2290  df-if 2372  df-pw 2412  df-sn 2422  df-pr 2423  df-tp 2425  df-op 2426  df-uni 2516  df-int 2546  df-iun 2580  df-br 2633  df-opab 2680  df-tr 2694  df-eprel 2846  df-id 2849  df-po 2854  df-so 2864  df-fr 2931  df-we 2948  df-ord 2965  df-on 2966  df-lim 2967  df-suc 2968  df-om 3146  df-xp 3198  df-rel 3199  df-cnv 3200  df-co 3201  df-dm 3202  df-rn 3203  df-res 3204  df-ima 3205  df-fun 3206  df-fn 3207  df-f 3208  df-f1 3209  df-fo 3210  df-f1o 3211  df-fv 3212  df-rdg 3946  df-opr 3979  df-oprab 3980  df-1st 4093  df-2nd 4094  df-1o 4147  df-oadd 4149  df-omul 4150  df-er 4275  df-ec 4277  df-qs 4280  df-en 4382  df-dom 4383  df-sdom 4384  df-ni 5013  df-pli 5014  df-mi 5015  df-lti 5016  df-plpq 5048  df-mpq 5049  df-enq 5050  df-nq 5051  df-plq 5052  df-mq 5053  df-rq 5054  df-ltq 5055  df-1q 5056  df-np 5099  df-1p 5100  df-plp 5101  df-mp 5102  df-ltp 5103  df-plpr 5177  df-mpr 5178  df-enr 5179  df-nr 5180  df-plr 5181  df-mr 5182  df-ltr 5183  df-0r 5184  df-1r 5185  df-m1r 5186  df-c 5253  df-0 5254  df-1 5255  df-i 5256  df-r 5257  df-plus 5258  df-mul 5259  df-lt 5260  df-sub 5369  df-neg 5371  df-pnf 5500  df-mnf 5501  df-xr 5502  df-ltxr 5503  df-le 5504  df-div 5716

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