Theorem dvdsrpropdg 13643

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 2202	. . . . . . 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 2491	. . . . 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 2263	. . . . 5 |- ( ph -> ( y e. B <-> y e. ( Base ` K ) ) )
9	7	rexeqdv 2697	. . . . 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 2263	. . . . 5 |- ( ph -> ( y e. B <-> y e. ( Base ` L ) ) )
13	11	rexeqdv 2697	. . . . 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 4095	. 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 2194	. . 3 |- ( ph -> ( Base ` K ) = ( Base ` K ) )
18		eqidd 2194	. . 3 |- ( ph -> ( ||r ` K ) = ( ||r ` K ) )
19		dvdsrpropdg.k	. . 3 |- ( ph -> K e. SRing )
20		eqidd 2194	. . 3 |- ( ph -> ( .r ` K ) = ( .r ` K ) )
21	17, 18, 19, 20	dvdsrvald 13589	. 2 |- ( ph -> ( ||r ` K ) = { <. y , z >. | ( y e. ( Base ` K ) /\ E. x e. ( Base ` K ) ( x ( .r ` K ) y ) = z ) } )
22		eqidd 2194	. . 3 |- ( ph -> ( Base ` L ) = ( Base ` L ) )
23		eqidd 2194	. . 3 |- ( ph -> ( ||r ` L ) = ( ||r ` L ) )
24		dvdsrpropdg.l	. . 3 |- ( ph -> L e. SRing )
25		eqidd 2194	. . 3 |- ( ph -> ( .r ` L ) = ( .r ` L ) )
26	22, 23, 24, 25	dvdsrvald 13589	. 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 2236	1 |- ( ph -> ( ||r ` K ) = ( ||r ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   E.wrex 2473   {copab 4089   ` cfv 5254  (class class class)co 5918   Basecbs 12618   .rcmulr 12696  SRingcsrg 13459   ||_rcdsr 13582

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-plusg 12708  df-mulr 12709  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-mgp 13417  df-srg 13460  df-dvdsr 13585

This theorem is referenced by:  unitpropdg  13644