Theorem pjthlem10 9228

Description: Lemma for pjth 9233.

Hypotheses

Ref	Expression
pjthlem9.1	|- A e. H~
pjthlem9.2	|- B e. H~
pjthlem9.3	|- D e. H~
pjthlem9.4	|- R = (1 / (D .ih D))
pjthlem9.5	|- S = (R x. (C .ih D))
pjthlem9.6	|- C = (A -h B)

Assertion

Ref	Expression
pjthlem10	|- ((D =/= 0h /\ (normh` (B -h A)) <_ (normh` ((B +h (S .h D)) -h A))) -> (R x. ((abs` (C .ih D))^2)) = 0)

Proof of Theorem pjthlem10

Step	Hyp	Ref	Expression
1		mulge0t 5679	. . . . 5 |- (((R e. RR /\ ((abs` (C .ih D))^2) e. RR) /\ (0 <_ R /\ 0 <_ ((abs` (C .ih D))^2))) -> 0 <_ (R x. ((abs` (C .ih D))^2)))
2		pjthlem9.3	. . . . . . 7 |- D e. H~
3		pjthlem9.4	. . . . . . 7 |- R = (1 / (D .ih D))
4	2, 3	pjthlem2 9220	. . . . . 6 |- (D =/= 0h -> R e. RR)
5		pjthlem9.6	. . . . . . . . . 10 |- C = (A -h B)
6		pjthlem9.1	. . . . . . . . . . 11 |- A e. H~
7		pjthlem9.2	. . . . . . . . . . 11 |- B e. H~
8	6, 7	hvsubcl 8891	. . . . . . . . . 10 |- (A -h B) e. H~
9	5, 8	eqeltr 1544	. . . . . . . . 9 |- C e. H~
10	9, 2	hicl 8948	. . . . . . . 8 |- (C .ih D) e. CC
11	10	abscl 6839	. . . . . . 7 |- (abs` (C .ih D)) e. RR
12	11	resqcl 6623	. . . . . 6 |- ((abs` (C .ih D))^2) e. RR
13	4, 12	jctir 293	. . . . 5 |- (D =/= 0h -> (R e. RR /\ ((abs` (C .ih D))^2) e. RR))
14		ltlet 5520	. . . . . . 7 |- ((0 e. RR /\ R e. RR) -> (0 < R -> 0 <_ R))
15		0re 5440	. . . . . . . 8 |- 0 e. RR
16	4, 15	jctil 292	. . . . . . 7 |- (D =/= 0h -> (0 e. RR /\ R e. RR))
17	2, 3	pjthlem3 9221	. . . . . . 7 |- (D =/= 0h -> 0 < R)
18	14, 16, 17	sylc 68	. . . . . 6 |- (D =/= 0h -> 0 <_ R)
19	11	sqge0 6628	. . . . . 6 |- 0 <_ ((abs` (C .ih D))^2)
20	18, 19	jctir 293	. . . . 5 |- (D =/= 0h -> (0 <_ R /\ 0 <_ ((abs` (C .ih D))^2)))
21	1, 13, 20	sylanc 471	. . . 4 |- (D =/= 0h -> 0 <_ (R x. ((abs` (C .ih D))^2)))
22	21	adantr 389	. . 3 |- ((D =/= 0h /\ (normh` (B -h A)) <_ (normh` ((B +h (S .h D)) -h A))) -> 0 <_ (R x. ((abs` (C .ih D))^2)))
23		pjthlem9.5	. . . . . 6 |- S = (R x. (C .ih D))
24	6, 7, 2, 3, 23, 5	pjthlem9 9227	. . . . 5 |- (D =/= 0h -> ((normh` (B -h A)) <_ (normh` ((B +h (S .h D)) -h A)) <-> ((normh` C)^2) <_ ((normh` (C -h (S .h D)))^2)))
25	24	biimpa 416	. . . 4 |- ((D =/= 0h /\ (normh` (B -h A)) <_ (normh` ((B +h (S .h D)) -h A))) -> ((normh` C)^2) <_ ((normh` (C -h (S .h D)))^2))
26	2, 3, 9, 23	pjthlem8 9226	. . . . 5 |- (D =/= 0h -> ((normh` (C -h (S .h D)))^2) = (((normh` C)^2) - (R x. ((abs` (C .ih D))^2))))
27	26	adantr 389	. . . 4 |- ((D =/= 0h /\ (normh` (B -h A)) <_ (normh` ((B +h (S .h D)) -h A))) -> ((normh` (C -h (S .h D)))^2) = (((normh` C)^2) - (R x. ((abs` (C .ih D))^2))))
28	25, 27	breqtrd 2639	. . 3 |- ((D =/= 0h /\ (normh` (B -h A)) <_ (normh` ((B +h (S .h D)) -h A))) -> ((normh` C)^2) <_ (((normh` C)^2) - (R x. ((abs` (C .ih D))^2))))
29	22, 28	jca 288	. 2 |- ((D =/= 0h /\ (normh` (B -h A)) <_ (normh` ((B +h (S .h D)) -h A))) -> (0 <_ (R x. ((abs` (C .ih D))^2)) /\ ((normh` C)^2) <_ (((normh` C)^2) - (R x. ((abs` (C .ih D))^2)))))
30		axmulrcl 5274	. . . . . 6 |- ((R e. RR /\ ((abs` (C .ih D))^2) e. RR) -> (R x. ((abs` (C .ih D))^2)) e. RR)
31	13, 30	syl 10	. . . . 5 |- (D =/= 0h -> (R x. ((abs` (C .ih D))^2)) e. RR)
32	9	normcl 8998	. . . . . 6 |- (normh` C) e. RR
33	32	resqcl 6623	. . . . 5 |- ((normh` C)^2) e. RR
34	31, 33	jctir 293	. . . 4 |- (D =/= 0h -> ((R x. ((abs` (C .ih D))^2)) e. RR /\ ((normh` C)^2) e. RR))
35		lesub0t 5678	. . . 4 |- (((R x. ((abs` (C .ih D))^2)) e. RR /\ ((normh` C)^2) e. RR) -> ((0 <_ (R x. ((abs` (C .ih D))^2)) /\ ((normh` C)^2) <_ (((normh` C)^2) - (R x. ((abs` (C .ih D))^2)))) <-> (R x. ((abs` (C .ih D))^2)) = 0))
36	34, 35	syl 10	. . 3 |- (D =/= 0h -> ((0 <_ (R x. ((abs` (C .ih D))^2)) /\ ((normh` C)^2) <_ (((normh` C)^2) - (R x. ((abs` (C .ih D))^2)))) <-> (R x. ((abs` (C .ih D))^2)) = 0))
37	36	adantr 389	. 2 |- ((D =/= 0h /\ (normh` (B -h A)) <_ (normh` ((B +h (S .h D)) -h A))) -> ((0 <_ (R x. ((abs` (C .ih D))^2)) /\ ((normh` C)^2) <_ (((normh` C)^2) - (R x. ((abs` (C .ih D))^2)))) <-> (R x. ((abs` (C .ih D))^2)) = 0))
38	29, 37	mpbid 195	1 |- ((D =/= 0h /\ (normh` (B -h A)) <_ (normh` ((B +h (S .h D)) -h A))) -> (R x. ((abs` (C .ih D))^2)) = 0)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958   =/= wne 1585   class class class wbr 2619  ` cfv 3182  (class class class)co 3963  RRcr 5233  0cc0 5234  1c1 5235   x. cmul 5239   - cmin 5292   / cdiv 5294   <_ cle 5295   < clt 5486  2c2 5961  ^cexp 6568  abscabs 6750  H~chil 8788   +h cva 8789   .h csm 8790  0hc0v 8791   -h cmv 8792   .ih csp 8793  normhcno 8794

This theorem is referenced by:  pjthlem11 9229

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625  ax-hfvadd 8870  ax-hvcom 8871  ax-hvass 8872  ax-hv0cl 8873  ax-hfvmul 8875  ax-hvmulid 8876  ax-hvmulass 8877  ax-hvdistr1 8878  ax-hvmul0 8880  ax-hfi 8946  ax-his1 8949  ax-his2 8950  ax-his3 8951  ax-his4 8952

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-en 4368  df-dom 4369  df-sdom 4370  df-sup 4574  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 5166  df-nr 5167  df-plr 5168  df-mr 5169  df-ltr 5170  df-0r 5171  df-1r 5172  df-m1r 5173  df-c 5240  df-0 5241  df-1 5242  df-i 5243  df-r 5244  df-plus 5245  df-mul 5246  df-lt 5247  df-sub 5356  df-neg 5358  df-pnf 5487  df-mnf 5488  df-xr 5489  df-ltxr 5490  df-le 5491  df-div 5703  df-n 5925  df-2 5970  df-n0 6100  df-z 6136  df-seq1 6308  df-exp 6569  df-sqr 6670  df-re 6751  df-im 6752  df-cj 6753  df-abs 6754  df-hnorm 8837  df-hvsub 8840

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