Theorem cdj3lem2 25839

Description: Lemma for cdj3i 25845. 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 24767	. . . 4 |- ( ( C e. A /\ D e. B ) -> ( C +h D ) e. ( A +H B ) )
4		eqeq1 2449	. . . . . . 7 |- ( x = ( C +h D ) -> ( x = ( z +h w ) <-> ( C +h D ) = ( z +h w ) ) )
5	4	rexbidv 2736	. . . . . 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 6054	. . . . 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 6056	. . . . 5 |- ( iota_ z e. A E. w e. B ( C +h D ) = ( z +h w ) ) e. _V
9	6, 7, 8	fvmpt 5774	. . . 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 16	. . 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 1008	. 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 2443	. . . . 5 |- ( C +h D ) = ( C +h D )
13		oveq2 6099	. . . . . . 7 |- ( w = D -> ( C +h w ) = ( C +h D ) )
14	13	eqeq2d 2454	. . . . . 6 |- ( w = D -> ( ( C +h D ) = ( C +h w ) <-> ( C +h D ) = ( C +h D ) ) )
15	14	rspcev 3073	. . . . 5 |- ( ( D e. B /\ ( C +h D ) = ( C +h D ) ) -> E. w e. B ( C +h D ) = ( C +h w ) )
16	12, 15	mpan2 671	. . . 4 |- ( D e. B -> E. w e. B ( C +h D ) = ( C +h w ) )
17	16	3ad2ant2 1010	. . 3 |- ( ( C e. A /\ D e. B /\ ( A i^i B ) = 0H ) -> E. w e. B ( C +h D ) = ( C +h w ) )
18		simp1 988	. . . 4 |- ( ( C e. A /\ D e. B /\ ( A i^i B ) = 0H ) -> C e. A )
19	1, 2	cdjreui 25836	. . . . . 6 |- ( ( ( 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	sylan 471	. . . . 5 |- ( ( ( 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	20	3impa 1182	. . . 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 ) )
22		oveq1 6098	. . . . . . 7 |- ( z = C -> ( z +h w ) = ( C +h w ) )
23	22	eqeq2d 2454	. . . . . 6 |- ( z = C -> ( ( C +h D ) = ( z +h w ) <-> ( C +h D ) = ( C +h w ) ) )
24	23	rexbidv 2736	. . . . 5 |- ( z = C -> ( E. w e. B ( C +h D ) = ( z +h w ) <-> E. w e. B ( C +h D ) = ( C +h w ) ) )
25	24	riota2 6075	. . . 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 ) )
26	18, 21, 25	syl2anc 661	. . 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 ) )
27	17, 26	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 )
28	11, 27	eqtrd 2475	1 |- ( ( C e. A /\ D e. B /\ ( A i^i B ) = 0H ) -> ( S ` ( C +h D ) ) = C )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   E.wrex 2716   E!wreu 2717   i^i cin 3327   e. cmpt 4350   ` cfv 5418   iota_crio 6051  (class class class)co 6091   +h cva 24322   SHcsh 24330   +H cph 24333   0Hc0h 24337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-hilex 24401  ax-hfvadd 24402  ax-hvcom 24403  ax-hvass 24404  ax-hv0cl 24405  ax-hvaddid 24406  ax-hfvmul 24407  ax-hvmulid 24408  ax-hvmulass 24409  ax-hvdistr1 24410  ax-hvdistr2 24411  ax-hvmul0 24412

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-grpo 23678  df-ablo 23769  df-hvsub 24373  df-sh 24609  df-ch0 24656  df-shs 24711

This theorem is referenced by:  cdj3lem2a  25840  cdj3lem2b  25841  cdj3lem3  25842  cdj3i  25845