Theorem dvdsrpropdg 13703

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 2205	. . . . . . 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 2494	. . . . 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 2266	. . . . 5 |- ( ph -> ( y e. B <-> y e. ( Base ` K ) ) )
9	7	rexeqdv 2700	. . . . 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 2266	. . . . 5 |- ( ph -> ( y e. B <-> y e. ( Base ` L ) ) )
13	11	rexeqdv 2700	. . . . 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 4099	. 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 2197	. . 3 |- ( ph -> ( Base ` K ) = ( Base ` K ) )
18		eqidd 2197	. . 3 |- ( ph -> ( ||r ` K ) = ( ||r ` K ) )
19		dvdsrpropdg.k	. . 3 |- ( ph -> K e. SRing )
20		eqidd 2197	. . 3 |- ( ph -> ( .r ` K ) = ( .r ` K ) )
21	17, 18, 19, 20	dvdsrvald 13649	. 2 |- ( ph -> ( ||r ` K ) = { <. y , z >. | ( y e. ( Base ` K ) /\ E. x e. ( Base ` K ) ( x ( .r ` K ) y ) = z ) } )
22		eqidd 2197	. . 3 |- ( ph -> ( Base ` L ) = ( Base ` L ) )
23		eqidd 2197	. . 3 |- ( ph -> ( ||r ` L ) = ( ||r ` L ) )
24		dvdsrpropdg.l	. . 3 |- ( ph -> L e. SRing )
25		eqidd 2197	. . 3 |- ( ph -> ( .r ` L ) = ( .r ` L ) )
26	22, 23, 24, 25	dvdsrvald 13649	. 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 2239	1 |- ( ph -> ( ||r ` K ) = ( ||r ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   E.wrex 2476   {copab 4093   ` cfv 5258  (class class class)co 5922   Basecbs 12678   .rcmulr 12756  SRingcsrg 13519   ||_rcdsr 13642

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-plusg 12768  df-mulr 12769  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-mgp 13477  df-srg 13520  df-dvdsr 13645

This theorem is referenced by:  unitpropdg  13704