Theorem cdj3lem2 27030

Description: Lemma for cdj3i 27036. 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 25958	. . . 4 |- ( ( C e. A /\ D e. B ) -> ( C +h D ) e. ( A +H B ) )
4		eqeq1 2471	. . . . . . 7 |- ( x = ( C +h D ) -> ( x = ( z +h w ) <-> ( C +h D ) = ( z +h w ) ) )
5	4	rexbidv 2973	. . . . . 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 6245	. . . . 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 6247	. . . . 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 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 1016	. 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 2467	. . . . 5 |- ( C +h D ) = ( C +h D )
13		oveq2 6290	. . . . . . 7 |- ( w = D -> ( C +h w ) = ( C +h D ) )
14	13	eqeq2d 2481	. . . . . 6 |- ( w = D -> ( ( C +h D ) = ( C +h w ) <-> ( C +h D ) = ( C +h D ) ) )
15	14	rspcev 3214	. . . . 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 1018	. . 3 |- ( ( C e. A /\ D e. B /\ ( A i^i B ) = 0H ) -> E. w e. B ( C +h D ) = ( C +h w ) )
18		simp1 996	. . . 4 |- ( ( C e. A /\ D e. B /\ ( A i^i B ) = 0H ) -> C e. A )
19	1, 2	cdjreui 27027	. . . . . 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 1191	. . . 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 6289	. . . . . . 7 |- ( z = C -> ( z +h w ) = ( C +h w ) )
23	22	eqeq2d 2481	. . . . . 6 |- ( z = C -> ( ( C +h D ) = ( z +h w ) <-> ( C +h D ) = ( C +h w ) ) )
24	23	rexbidv 2973	. . . . 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 6266	. . . 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 2508	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 973   = wceq 1379   e. wcel 1767   E.wrex 2815   E!wreu 2816   i^i cin 3475   |-> cmpt 4505   ` cfv 5586   iota_crio 6242  (class class class)co 6282   +h cva 25513   SHcsh 25521   +H cph 25524   0Hc0h 25528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-hilex 25592  ax-hfvadd 25593  ax-hvcom 25594  ax-hvass 25595  ax-hv0cl 25596  ax-hvaddid 25597  ax-hfvmul 25598  ax-hvmulid 25599  ax-hvmulass 25600  ax-hvdistr1 25601  ax-hvdistr2 25602  ax-hvmul0 25603

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-grpo 24869  df-ablo 24960  df-hvsub 25564  df-sh 25800  df-ch0 25847  df-shs 25902

This theorem is referenced by:  cdj3lem2a  27031  cdj3lem2b  27032  cdj3lem3  27033  cdj3i  27036