Theorem cdj3lem2 27753

Description: Lemma for cdj3i 27759. 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 26682	. . . 4 |- ( ( C e. A /\ D e. B ) -> ( C +h D ) e. ( A +H B ) )
4		eqeq1 2406	. . . . . . 7 |- ( x = ( C +h D ) -> ( x = ( z +h w ) <-> ( C +h D ) = ( z +h w ) ) )
5	4	rexbidv 2917	. . . . . 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 6241	. . . . 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 6243	. . . . 5 |- ( iota_ z e. A E. w e. B ( C +h D ) = ( z +h w ) ) e. _V
9	6, 7, 8	fvmpt 5931	. . . 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 1017	. 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 2402	. . . . 5 |- ( C +h D ) = ( C +h D )
13		oveq2 6285	. . . . . . 7 |- ( w = D -> ( C +h w ) = ( C +h D ) )
14	13	eqeq2d 2416	. . . . . 6 |- ( w = D -> ( ( C +h D ) = ( C +h w ) <-> ( C +h D ) = ( C +h D ) ) )
15	14	rspcev 3159	. . . . 5 |- ( ( D e. B /\ ( C +h D ) = ( C +h D ) ) -> E. w e. B ( C +h D ) = ( C +h w ) )
16	12, 15	mpan2 669	. . . 4 |- ( D e. B -> E. w e. B ( C +h D ) = ( C +h w ) )
17	16	3ad2ant2 1019	. . 3 |- ( ( C e. A /\ D e. B /\ ( A i^i B ) = 0H ) -> E. w e. B ( C +h D ) = ( C +h w ) )
18		simp1 997	. . . 4 |- ( ( C e. A /\ D e. B /\ ( A i^i B ) = 0H ) -> C e. A )
19	1, 2	cdjreui 27750	. . . . 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 1630	. . . 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 6284	. . . . . . 7 |- ( z = C -> ( z +h w ) = ( C +h w ) )
22	21	eqeq2d 2416	. . . . . 6 |- ( z = C -> ( ( C +h D ) = ( z +h w ) <-> ( C +h D ) = ( C +h w ) ) )
23	22	rexbidv 2917	. . . . 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 6261	. . . 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 659	. . 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 210	. 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 2443	1 |- ( ( C e. A /\ D e. B /\ ( A i^i B ) = 0H ) -> ( S ` ( C +h D ) ) = C )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 974   = wceq 1405   e. wcel 1842   E.wrex 2754   E!wreu 2755   i^i cin 3412   |-> cmpt 4452   ` cfv 5568   iota_crio 6238  (class class class)co 6277   +h cva 26237   SHcsh 26245   +H cph 26248   0Hc0h 26252

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-hilex 26316  ax-hfvadd 26317  ax-hvcom 26318  ax-hvass 26319  ax-hv0cl 26320  ax-hvaddid 26321  ax-hfvmul 26322  ax-hvmulid 26323  ax-hvmulass 26324  ax-hvdistr1 26325  ax-hvdistr2 26326  ax-hvmul0 26327

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-po 4743  df-so 4744  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-grpo 25593  df-ablo 25684  df-hvsub 26288  df-sh 26524  df-ch0 26571  df-shs 26626

This theorem is referenced by:  cdj3lem2a  27754  cdj3lem2b  27755  cdj3lem3  27756  cdj3i  27759