Theorem pjthlem10 9166

Description: Lemma for pjth 9171.

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 5660	. . . . 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 9158	. . . . . 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 8830	. . . . . . . . . 10 |- (A -h B) e. H~
9	5, 8	eqeltr 1541	. . . . . . . . 9 |- C e. H~
10	9, 2	hicl 8887	. . . . . . . 8 |- (C .ih D) e. CC
11	10	abscl 6782	. . . . . . 7 |- (abs` (C .ih D)) e. RR
12	11	resqcl 6562	. . . . . 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 5501	. . . . . . 7 |- ((0 e. RR /\ R e. RR) -> (0 < R -> 0 <_ R))
15		0re 5420	. . . . . . . 8 |- 0 e. RR
16	4, 15	jctil 292	. . . . . . 7 |- (D =/= 0h -> (0 e. RR /\ R e. RR))
17	2, 3	pjthlem3 9159	. . . . . . 7 |- (D =/= 0h -> 0 < R)
18	14, 16, 17	sylc 68	. . . . . 6 |- (D =/= 0h -> 0 <_ R)
19	11	sqge0 6567	. . . . . 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 9165	. . . . 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 9164	. . . . 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 2634	. . 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 5254	. . . . . 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 8937	. . . . . 6 |- (normh` C) e. RR
33	32	resqcl 6562	. . . . 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 5659	. . . 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 954   e. wcel 956   =/= wne 1582   class class class wbr 2614  ` cfv 3177  (class class class)co 3954  RRcr 5213  0cc0 5214  1c1 5215   x. cmul 5219   - cmin 5272   / cdiv 5274   <_ cle 5275   < clt 5466  2c2 5916  ^cexp 6508  abscabs 6689  H~chil 8727   +h cva 8728   .h csm 8729  0hc0v 8730   -h cmv 8731   .ih csp 8732  normhcno 8733

This theorem is referenced by:  pjthlem11 9167

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605  ax-hfvadd 8809  ax-hvcom 8810  ax-hvass 8811  ax-hv0cl 8812  ax-hfvmul 8814  ax-hvmulid 8815  ax-hvmulass 8816  ax-hvdistr1 8817  ax-hvmul0 8819  ax-hfi 8885  ax-his1 8888  ax-his2 8889  ax-his3 8890  ax-his4 8891

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-nel 1585  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-1o 4123  df-oadd 4125  df-omul 4126  df-er 4251  df-ec 4253  df-qs 4256  df-en 4357  df-dom 4358  df-sdom 4359  df-sup 4554  df-ni 4980  df-pli 4981  df-mi 4982  df-lti 4983  df-plpq 5015  df-mpq 5016  df-enq 5017  df-nq 5018  df-plq 5019  df-mq 5020  df-rq 5021  df-ltq 5022  df-1q 5023  df-np 5066  df-1p 5067  df-plp 5068  df-mp 5069  df-ltp 5070  df-plpr 5144  df-mpr 5145  df-enr 5146  df-nr 5147  df-plr 5148  df-mr 5149  df-ltr 5150  df-0r 5151  df-1r 5152  df-m1r 5153  df-c 5220  df-0 5221  df-1 5222  df-i 5223  df-r 5224  df-plus 5225  df-mul 5226  df-lt 5227  df-sub 5336  df-neg 5338  df-pnf 5467  df-mnf 5468  df-xr 5469  df-ltxr 5470  df-le 5471  df-div 5680  df-n 5881  df-2 5925  df-n0 6055  df-z 6091  df-seq1 6253  df-exp 6509  df-sqr 6608  df-re 6690  df-im 6691  df-cj 6692  df-abs 6693  df-hnorm 8776  df-hvsub 8779

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