Theorem dvdsrpropdg 13909

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 2214	. . . . . . 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 2503	. . . . 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 2275	. . . . 5 |- ( ph -> ( y e. B <-> y e. ( Base ` K ) ) )
9	7	rexeqdv 2709	. . . . 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 2275	. . . . 5 |- ( ph -> ( y e. B <-> y e. ( Base ` L ) ) )
13	11	rexeqdv 2709	. . . . 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 4110	. 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 2206	. . 3 |- ( ph -> ( Base ` K ) = ( Base ` K ) )
18		eqidd 2206	. . 3 |- ( ph -> ( ||r ` K ) = ( ||r ` K ) )
19		dvdsrpropdg.k	. . 3 |- ( ph -> K e. SRing )
20		eqidd 2206	. . 3 |- ( ph -> ( .r ` K ) = ( .r ` K ) )
21	17, 18, 19, 20	dvdsrvald 13855	. 2 |- ( ph -> ( ||r ` K ) = { <. y , z >. | ( y e. ( Base ` K ) /\ E. x e. ( Base ` K ) ( x ( .r ` K ) y ) = z ) } )
22		eqidd 2206	. . 3 |- ( ph -> ( Base ` L ) = ( Base ` L ) )
23		eqidd 2206	. . 3 |- ( ph -> ( ||r ` L ) = ( ||r ` L ) )
24		dvdsrpropdg.l	. . 3 |- ( ph -> L e. SRing )
25		eqidd 2206	. . 3 |- ( ph -> ( .r ` L ) = ( .r ` L ) )
26	22, 23, 24, 25	dvdsrvald 13855	. 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 2248	1 |- ( ph -> ( ||r ` K ) = ( ||r ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2176   E.wrex 2485   {copab 4104   ` cfv 5271  (class class class)co 5944   Basecbs 12832   .rcmulr 12910  SRingcsrg 13725   ||_rcdsr 13848

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-pre-ltirr 8037  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-ltxr 8112  df-inn 9037  df-2 9095  df-3 9096  df-ndx 12835  df-slot 12836  df-base 12838  df-sets 12839  df-plusg 12922  df-mulr 12923  df-0g 13090  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-mgp 13683  df-srg 13726  df-dvdsr 13851

This theorem is referenced by:  unitpropdg  13910