Theorem osumlem2 9574

Description: Lemma for osum 9581.

Hypotheses

Ref	Expression
osumlem1.1	|- A e. CH
osumlem1.2	|- B e. CH
osumlem1.3	|- B (_ (_|_` A)
osumlem1.4	|- (ph <-> (((C e. A /\ D e. B) /\ R = (C +h D)) /\ ((x e. A /\ y e. (_|_` A)) /\ z = (x +h y))))

Assertion

Ref	Expression
osumlem2	|- (ph -> (((normh` (C -h x))^2) + ((normh` (D -h y))^2)) = ((normh` (R -h z))^2))

Proof of Theorem osumlem2

Step	Hyp	Ref	Expression
1		osumlem1.4	. 2 |- (ph <-> (((C e. A /\ D e. B) /\ R = (C +h D)) /\ ((x e. A /\ y e. (_|_` A)) /\ z = (x +h y))))
2		pm3.27 323	. . . . . . 7 |- (((C e. A /\ D e. B) /\ R = (C +h D)) -> R = (C +h D))
3		pm3.27 323	. . . . . . 7 |- (((x e. A /\ y e. (_|_` A)) /\ z = (x +h y)) -> z = (x +h y))
4	2, 3	opreqan12d 3985	. . . . . 6 |- ((((C e. A /\ D e. B) /\ R = (C +h D)) /\ ((x e. A /\ y e. (_|_` A)) /\ z = (x +h y))) -> (R -h z) = ((C +h D) -h (x +h y)))
5		osumlem1.1	. . . . . . . 8 |- A e. CH
6		osumlem1.2	. . . . . . . 8 |- B e. CH
7		osumlem1.3	. . . . . . . 8 |- B (_ (_|_` A)
8		pm4.2 170	. . . . . . . 8 |- ((((C e. A /\ D e. B) /\ R = (C +h D)) /\ ((x e. A /\ y e. (_|_` A)) /\ z = (x +h y))) <-> (((C e. A /\ D e. B) /\ R = (C +h D)) /\ ((x e. A /\ y e. (_|_` A)) /\ z = (x +h y))))
9	5, 6, 7, 8	osumlem1 9573	. . . . . . 7 |- ((((C e. A /\ D e. B) /\ R = (C +h D)) /\ ((x e. A /\ y e. (_|_` A)) /\ z = (x +h y))) -> (((C e. H~ /\ D e. H~) /\ R e. H~) /\ ((x e. H~ /\ y e. H~) /\ z e. H~)))
10		hvsub4t 8901	. . . . . . . 8 |- (((C e. H~ /\ D e. H~) /\ (x e. H~ /\ y e. H~)) -> ((C +h D) -h (x +h y)) = ((C -h x) +h (D -h y)))
11	10	ad2ant2r 411	. . . . . . 7 |- ((((C e. H~ /\ D e. H~) /\ R e. H~) /\ ((x e. H~ /\ y e. H~) /\ z e. H~)) -> ((C +h D) -h (x +h y)) = ((C -h x) +h (D -h y)))
12	9, 11	syl 10	. . . . . 6 |- ((((C e. A /\ D e. B) /\ R = (C +h D)) /\ ((x e. A /\ y e. (_|_` A)) /\ z = (x +h y))) -> ((C +h D) -h (x +h y)) = ((C -h x) +h (D -h y)))
13	4, 12	eqtrd 1510	. . . . 5 |- ((((C e. A /\ D e. B) /\ R = (C +h D)) /\ ((x e. A /\ y e. (_|_` A)) /\ z = (x +h y))) -> (R -h z) = ((C -h x) +h (D -h y)))
14	13	fveq2d 3734	. . . 4 |- ((((C e. A /\ D e. B) /\ R = (C +h D)) /\ ((x e. A /\ y e. (_|_` A)) /\ z = (x +h y))) -> (normh` (R -h z)) = (normh` ((C -h x) +h (D -h y))))
15	14	opreq1d 3981	. . 3 |- ((((C e. A /\ D e. B) /\ R = (C +h D)) /\ ((x e. A /\ y e. (_|_` A)) /\ z = (x +h y))) -> ((normh` (R -h z))^2) = ((normh` ((C -h x) +h (D -h y)))^2))
16		normpytht 9007	. . . 4 |- (((C -h x) e. H~ /\ (D -h y) e. H~) -> (((C -h x) .ih (D -h y)) = 0 -> ((normh` ((C -h x) +h (D -h y)))^2) = (((normh` (C -h x))^2) + ((normh` (D -h y))^2))))
17		hvsubclt 8882	. . . . . . . 8 |- ((C e. H~ /\ x e. H~) -> (C -h x) e. H~)
18		hvsubclt 8882	. . . . . . . 8 |- ((D e. H~ /\ y e. H~) -> (D -h y) e. H~)
19	17, 18	anim12i 333	. . . . . . 7 |- (((C e. H~ /\ x e. H~) /\ (D e. H~ /\ y e. H~)) -> ((C -h x) e. H~ /\ (D -h y) e. H~))
20	19	an4s 510	. . . . . 6 |- (((C e. H~ /\ D e. H~) /\ (x e. H~ /\ y e. H~)) -> ((C -h x) e. H~ /\ (D -h y) e. H~))
21	20	ad2ant2r 411	. . . . 5 |- ((((C e. H~ /\ D e. H~) /\ R e. H~) /\ ((x e. H~ /\ y e. H~) /\ z e. H~)) -> ((C -h x) e. H~ /\ (D -h y) e. H~))
22	9, 21	syl 10	. . . 4 |- ((((C e. A /\ D e. B) /\ R = (C +h D)) /\ ((x e. A /\ y e. (_|_` A)) /\ z = (x +h y))) -> ((C -h x) e. H~ /\ (D -h y) e. H~))
23	5	chshi 9092	. . . . . . . 8 |- A e. SH
24		shocorth 9160	. . . . . . . 8 |- (A e. SH -> (((C -h x) e. A /\ (D -h y) e. (_|_` A)) -> ((C -h x) .ih (D -h y)) = 0))
25	23, 24	ax-mp 7	. . . . . . 7 |- (((C -h x) e. A /\ (D -h y) e. (_|_` A)) -> ((C -h x) .ih (D -h y)) = 0)
26		shsubcltOLD 9085	. . . . . . . 8 |- (A e. SH -> ((C e. A /\ x e. A) -> (C -h x) e. A))
27	23, 26	ax-mp 7	. . . . . . 7 |- ((C e. A /\ x e. A) -> (C -h x) e. A)
28	5	choccl 9180	. . . . . . . . . 10 |- (_|_` A) e. CH
29	28	chshi 9092	. . . . . . . . 9 |- (_|_` A) e. SH
30		shsubcltOLD 9085	. . . . . . . . 9 |- ((_|_` A) e. SH -> ((D e. (_|_` A) /\ y e. (_|_` A)) -> (D -h y) e. (_|_` A)))
31	29, 30	ax-mp 7	. . . . . . . 8 |- ((D e. (_|_` A) /\ y e. (_|_` A)) -> (D -h y) e. (_|_` A))
32	7	sseli 2068	. . . . . . . 8 |- (D e. B -> D e. (_|_` A))
33	31, 32	sylan 450	. . . . . . 7 |- ((D e. B /\ y e. (_|_` A)) -> (D -h y) e. (_|_` A))
34	25, 27, 33	syl2an 456	. . . . . 6 |- (((C e. A /\ x e. A) /\ (D e. B /\ y e. (_|_` A))) -> ((C -h x) .ih (D -h y)) = 0)
35	34	an4s 510	. . . . 5 |- (((C e. A /\ D e. B) /\ (x e. A /\ y e. (_|_` A))) -> ((C -h x) .ih (D -h y)) = 0)
36	35	ad2ant2r 411	. . . 4 |- ((((C e. A /\ D e. B) /\ R = (C +h D)) /\ ((x e. A /\ y e. (_|_` A)) /\ z = (x +h y))) -> ((C -h x) .ih (D -h y)) = 0)
37	16, 22, 36	sylc 68	. . 3 |- ((((C e. A /\ D e. B) /\ R = (C +h D)) /\ ((x e. A /\ y e. (_|_` A)) /\ z = (x +h y))) -> ((normh` ((C -h x) +h (D -h y)))^2) = (((normh` (C -h x))^2) + ((normh` (D -h y))^2)))
38	15, 37	eqtr2d 1511	. 2 |- ((((C e. A /\ D e. B) /\ R = (C +h D)) /\ ((x e. A /\ y e. (_|_` A)) /\ z = (x +h y))) -> (((normh` (C -h x))^2) + ((normh` (D -h y))^2)) = ((normh` (R -h z))^2))
39	1, 38	sylbi 199	1 |- (ph -> (((normh` (C -h x))^2) + ((normh` (D -h y))^2)) = ((normh` (R -h z))^2))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960   (_ wss 2050  ` cfv 3188  (class class class)co 3969  0cc0 5246   + caddc 5249  2c2 5963  ^cexp 6569  H~chil 8783   +h cva 8784   -h cmv 8787   .ih csp 8788  normhcno 8789  SHcsh 8792  CHcch 8793  _|_cort 8794

This theorem is referenced by:  osumlem3 9575

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-reg 4602  ax-inf2 4634  ax-ac 4754  ax-hilex 8864  ax-hfvadd 8865  ax-hvcom 8866  ax-hvass 8867  ax-hv0cl 8868  ax-hvaddid 8869  ax-hfvmul 8870  ax-hvmulid 8871  ax-hvmulass 8872  ax-hvdistr1 8873  ax-hvdistr2 8874  ax-hvmul0 8875  ax-hfi 8941  ax-his1 8944  ax-his2 8945  ax-his3 8946  ax-his4 8947

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-iin 2573  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846