Theorem dvdsrpropdg 14392

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 2243	. . . . . . 7 |- ( ( ( ph /\ x e. B ) /\ y e. B ) -> ( ( x ( .r ` K ) y ) = z <-> ( x ( .r ` L ) y ) = z ) )
4	3	an32s 570	. . . . . 6 |- ( ( ( ph /\ y e. B ) /\ x e. B ) -> ( ( x ( .r ` K ) y ) = z <-> ( x ( .r ` L ) y ) = z ) )
5	4	rexbidva 2541	. . . . 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 2304	. . . . 5 |- ( ph -> ( y e. B <-> y e. ( Base ` K ) ) )
9	7	rexeqdv 2750	. . . . 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 2304	. . . . 5 |- ( ph -> ( y e. B <-> y e. ( Base ` L ) ) )
13	11	rexeqdv 2750	. . . . 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 4181	. 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 2235	. . 3 |- ( ph -> ( Base ` K ) = ( Base ` K ) )
18		eqidd 2235	. . 3 |- ( ph -> ( ||r ` K ) = ( ||r ` K ) )
19		dvdsrpropdg.k	. . 3 |- ( ph -> K e. SRing )
20		eqidd 2235	. . 3 |- ( ph -> ( .r ` K ) = ( .r ` K ) )
21	17, 18, 19, 20	dvdsrvald 14338	. 2 |- ( ph -> ( ||r ` K ) = { <. y , z >. | ( y e. ( Base ` K ) /\ E. x e. ( Base ` K ) ( x ( .r ` K ) y ) = z ) } )
22		eqidd 2235	. . 3 |- ( ph -> ( Base ` L ) = ( Base ` L ) )
23		eqidd 2235	. . 3 |- ( ph -> ( ||r ` L ) = ( ||r ` L ) )
24		dvdsrpropdg.l	. . 3 |- ( ph -> L e. SRing )
25		eqidd 2235	. . 3 |- ( ph -> ( .r ` L ) = ( .r ` L ) )
26	22, 23, 24, 25	dvdsrvald 14338	. 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 2277	1 |- ( ph -> ( ||r ` K ) = ( ||r ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   E.wrex 2523   {copab 4175   ` cfv 5357  (class class class)co 6058   Basecbs 13296   .rcmulr 13375  SRingcsrg 14206   ||_rcdsr 14330

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-plusg 13387  df-mulr 13388  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-mgp 14160  df-srg 14207  df-dvdsr 14333

This theorem is referenced by:  unitpropdg  14393