Theorem dvdsrpropdg 14151

Description: The divisibility relation depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.)

Hypotheses

Ref	Expression
dvdsrpropdg.1	|- ( ph -> B = ( Base ` K ) )
dvdsrpropdg.2	|- ( ph -> B = ( Base ` L ) )
dvdsrpropdg.3	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )
dvdsrpropdg.k	|- ( ph -> K e. SRing )
dvdsrpropdg.l	|- ( ph -> L e. SRing )

Assertion

Ref	Expression
dvdsrpropdg	|- ( ph -> ( ||r ` K ) = ( ||r ` L ) )

Distinct variable groups:   x, y, B   x, K, y   x, L, y   ph, x, y

Proof of Theorem dvdsrpropdg

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvdsrpropdg.3	. . . . . . . . 9 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )
2	1	anassrs 400	. . . . . . . 8 |- ( ( ( ph /\ x e. B ) /\ y e. B ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )
3	2	eqeq1d 2238	. . . . . . 7 |- ( ( ( ph /\ x e. B ) /\ y e. B ) -> ( ( x ( .r ` K ) y ) = z <-> ( x ( .r ` L ) y ) = z ) )
4	3	an32s 568	. . . . . 6 |- ( ( ( ph /\ y e. B ) /\ x e. B ) -> ( ( x ( .r ` K ) y ) = z <-> ( x ( .r ` L ) y ) = z ) )
5	4	rexbidva 2527	. . . . 5 |- ( ( ph /\ y e. B ) -> ( E. x e. B ( x ( .r ` K ) y ) = z <-> E. x e. B ( x ( .r ` L ) y ) = z ) )
6	5	pm5.32da 452	. . . 4 |- ( ph -> ( ( y e. B /\ E. x e. B ( x ( .r ` K ) y ) = z ) <-> ( y e. B /\ E. x e. B ( x ( .r ` L ) y ) = z ) ) )
7		dvdsrpropdg.1	. . . . . 6 |- ( ph -> B = ( Base ` K ) )
8	7	eleq2d 2299	. . . . 5 |- ( ph -> ( y e. B <-> y e. ( Base ` K ) ) )
9	7	rexeqdv 2735	. . . . 5 |- ( ph -> ( E. x e. B ( x ( .r ` K ) y ) = z <-> E. x e. ( Base ` K ) ( x ( .r ` K ) y ) = z ) )
10	8, 9	anbi12d 473	. . . 4 |- ( ph -> ( ( y e. B /\ E. x e. B ( x ( .r ` K ) y ) = z ) <-> ( y e. ( Base ` K ) /\ E. x e. ( Base ` K ) ( x ( .r ` K ) y ) = z ) ) )
11		dvdsrpropdg.2	. . . . . 6 |- ( ph -> B = ( Base ` L ) )
12	11	eleq2d 2299	. . . . 5 |- ( ph -> ( y e. B <-> y e. ( Base ` L ) ) )
13	11	rexeqdv 2735	. . . . 5 |- ( ph -> ( E. x e. B ( x ( .r ` L ) y ) = z <-> E. x e. ( Base ` L ) ( x ( .r ` L ) y ) = z ) )
14	12, 13	anbi12d 473	. . . 4 |- ( ph -> ( ( y e. B /\ E. x e. B ( x ( .r ` L ) y ) = z ) <-> ( y e. ( Base ` L ) /\ E. x e. ( Base ` L ) ( x ( .r ` L ) y ) = z ) ) )
15	6, 10, 14	3bitr3d 218	. . 3 |- ( ph -> ( ( y e. ( Base ` K ) /\ E. x e. ( Base ` K ) ( x ( .r ` K ) y ) = z ) <-> ( y e. ( Base ` L ) /\ E. x e. ( Base ` L ) ( x ( .r ` L ) y ) = z ) ) )
16	15	opabbidv 4153	. 2 |- ( ph -> { <. y , z >. | ( y e. ( Base ` K ) /\ E. x e. ( Base ` K ) ( x ( .r ` K ) y ) = z ) } = { <. y , z >. | ( y e. ( Base ` L ) /\ E. x e. ( Base ` L ) ( x ( .r ` L ) y ) = z ) } )
17		eqidd 2230	. . 3 |- ( ph -> ( Base ` K ) = ( Base ` K ) )
18		eqidd 2230	. . 3 |- ( ph -> ( ||r ` K ) = ( ||r ` K ) )
19		dvdsrpropdg.k	. . 3 |- ( ph -> K e. SRing )
20		eqidd 2230	. . 3 |- ( ph -> ( .r ` K ) = ( .r ` K ) )
21	17, 18, 19, 20	dvdsrvald 14097	. 2 |- ( ph -> ( ||r ` K ) = { <. y , z >. | ( y e. ( Base ` K ) /\ E. x e. ( Base ` K ) ( x ( .r ` K ) y ) = z ) } )
22		eqidd 2230	. . 3 |- ( ph -> ( Base ` L ) = ( Base ` L ) )
23		eqidd 2230	. . 3 |- ( ph -> ( ||r ` L ) = ( ||r ` L ) )
24		dvdsrpropdg.l	. . 3 |- ( ph -> L e. SRing )
25		eqidd 2230	. . 3 |- ( ph -> ( .r ` L ) = ( .r ` L ) )
26	22, 23, 24, 25	dvdsrvald 14097	. 2 |- ( ph -> ( ||r ` L ) = { <. y , z >. | ( y e. ( Base ` L ) /\ E. x e. ( Base ` L ) ( x ( .r ` L ) y ) = z ) } )
27	16, 21, 26	3eqtr4d 2272	1 |- ( ph -> ( ||r ` K ) = ( ||r ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   E.wrex 2509   {copab 4147   ` cfv 5324  (class class class)co 6013   Basecbs 13072   .rcmulr 13151  SRingcsrg 13966   ||_rcdsr 14089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-pre-ltirr 8134  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-iota 5284  df-fun 5326  df-fn 5327  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8206  df-mnf 8207  df-ltxr 8209  df-inn 9134  df-2 9192  df-3 9193  df-ndx 13075  df-slot 13076  df-base 13078  df-sets 13079  df-plusg 13163  df-mulr 13164  df-0g 13331  df-mgm 13429  df-sgrp 13475  df-mnd 13490  df-mgp 13924  df-srg 13967  df-dvdsr 14092

This theorem is referenced by:  unitpropdg  14152