Theorem cdj3lem2 28088

Description: Lemma for cdj3i 28094. Value of the first-component function S. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdj3lem2.1	|- A e. SH
cdj3lem2.2	|- B e. SH
cdj3lem2.3	|- S = ( x e. ( A +H B ) |-> ( iota_ z e. A E. w e. B x = ( z +h w ) ) )

Assertion

Ref	Expression
cdj3lem2	|- ( ( C e. A /\ D e. B /\ ( A i^i B ) = 0H ) -> ( S ` ( C +h D ) ) = C )

Distinct variable groups:   x, z, w, A   x, B, z, w   x, C, z, w   x, D, z, w

Allowed substitution hints:   S( x, z, w)

Proof of Theorem cdj3lem2

Step	Hyp	Ref	Expression
1		cdj3lem2.1	. . . . 5 |- A e. SH
2		cdj3lem2.2	. . . . 5 |- B e. SH
3	1, 2	shsvai 27017	. . . 4 |- ( ( C e. A /\ D e. B ) -> ( C +h D ) e. ( A +H B ) )
4		eqeq1 2455	. . . . . . 7 |- ( x = ( C +h D ) -> ( x = ( z +h w ) <-> ( C +h D ) = ( z +h w ) ) )
5	4	rexbidv 2901	. . . . . 6 |- ( x = ( C +h D ) -> ( E. w e. B x = ( z +h w ) <-> E. w e. B ( C +h D ) = ( z +h w ) ) )
6	5	riotabidv 6254	. . . . 5 |- ( x = ( C +h D ) -> ( iota_ z e. A E. w e. B x = ( z +h w ) ) = ( iota_ z e. A E. w e. B ( C +h D ) = ( z +h w ) ) )
7		cdj3lem2.3	. . . . 5 |- S = ( x e. ( A +H B ) |-> ( iota_ z e. A E. w e. B x = ( z +h w ) ) )
8		riotaex 6256	. . . . 5 |- ( iota_ z e. A E. w e. B ( C +h D ) = ( z +h w ) ) e. _V
9	6, 7, 8	fvmpt 5948	. . . 4 |- ( ( C +h D ) e. ( A +H B ) -> ( S ` ( C +h D ) ) = ( iota_ z e. A E. w e. B ( C +h D ) = ( z +h w ) ) )
10	3, 9	syl 17	. . 3 |- ( ( C e. A /\ D e. B ) -> ( S ` ( C +h D ) ) = ( iota_ z e. A E. w e. B ( C +h D ) = ( z +h w ) ) )
11	10	3adant3 1028	. 2 |- ( ( C e. A /\ D e. B /\ ( A i^i B ) = 0H ) -> ( S ` ( C +h D ) ) = ( iota_ z e. A E. w e. B ( C +h D ) = ( z +h w ) ) )
12		eqid 2451	. . . . 5 |- ( C +h D ) = ( C +h D )
13		oveq2 6298	. . . . . . 7 |- ( w = D -> ( C +h w ) = ( C +h D ) )
14	13	eqeq2d 2461	. . . . . 6 |- ( w = D -> ( ( C +h D ) = ( C +h w ) <-> ( C +h D ) = ( C +h D ) ) )
15	14	rspcev 3150	. . . . 5 |- ( ( D e. B /\ ( C +h D ) = ( C +h D ) ) -> E. w e. B ( C +h D ) = ( C +h w ) )
16	12, 15	mpan2 677	. . . 4 |- ( D e. B -> E. w e. B ( C +h D ) = ( C +h w ) )
17	16	3ad2ant2 1030	. . 3 |- ( ( C e. A /\ D e. B /\ ( A i^i B ) = 0H ) -> E. w e. B ( C +h D ) = ( C +h w ) )
18		simp1 1008	. . . 4 |- ( ( C e. A /\ D e. B /\ ( A i^i B ) = 0H ) -> C e. A )
19	1, 2	cdjreui 28085	. . . . 5 |- ( ( ( C +h D ) e. ( A +H B ) /\ ( A i^i B ) = 0H ) -> E! z e. A E. w e. B ( C +h D ) = ( z +h w ) )
20	3, 19	stoic3 1660	. . . 4 |- ( ( C e. A /\ D e. B /\ ( A i^i B ) = 0H ) -> E! z e. A E. w e. B ( C +h D ) = ( z +h w ) )
21		oveq1 6297	. . . . . . 7 |- ( z = C -> ( z +h w ) = ( C +h w ) )
22	21	eqeq2d 2461	. . . . . 6 |- ( z = C -> ( ( C +h D ) = ( z +h w ) <-> ( C +h D ) = ( C +h w ) ) )
23	22	rexbidv 2901	. . . . 5 |- ( z = C -> ( E. w e. B ( C +h D ) = ( z +h w ) <-> E. w e. B ( C +h D ) = ( C +h w ) ) )
24	23	riota2 6274	. . . 4 |- ( ( C e. A /\ E! z e. A E. w e. B ( C +h D ) = ( z +h w ) ) -> ( E. w e. B ( C +h D ) = ( C +h w ) <-> ( iota_ z e. A E. w e. B ( C +h D ) = ( z +h w ) ) = C ) )
25	18, 20, 24	syl2anc 667	. . 3 |- ( ( C e. A /\ D e. B /\ ( A i^i B ) = 0H ) -> ( E. w e. B ( C +h D ) = ( C +h w ) <-> ( iota_ z e. A E. w e. B ( C +h D ) = ( z +h w ) ) = C ) )
26	17, 25	mpbid 214	. 2 |- ( ( C e. A /\ D e. B /\ ( A i^i B ) = 0H ) -> ( iota_ z e. A E. w e. B ( C +h D ) = ( z +h w ) ) = C )
27	11, 26	eqtrd 2485	1 |- ( ( C e. A /\ D e. B /\ ( A i^i B ) = 0H ) -> ( S ` ( C +h D ) ) = C )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   E.wrex 2738   E!wreu 2739   i^i cin 3403   |-> cmpt 4461   ` cfv 5582   iota_crio 6251  (class class class)co 6290   +h cva 26573   SHcsh 26581   +H cph 26584   0Hc0h 26588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-hilex 26652  ax-hfvadd 26653  ax-hvcom 26654  ax-hvass 26655  ax-hv0cl 26656  ax-hvaddid 26657  ax-hfvmul 26658  ax-hvmulid 26659  ax-hvmulass 26660  ax-hvdistr1 26661  ax-hvdistr2 26662  ax-hvmul0 26663

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-po 4755  df-so 4756  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-grpo 25919  df-ablo 26010  df-hvsub 26624  df-sh 26860  df-ch0 26906  df-shs 26961

This theorem is referenced by:  cdj3lem2a  28089  cdj3lem2b  28090  cdj3lem3  28091  cdj3i  28094