Theorem rngidpropdg 13320

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

Hypotheses

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

Assertion

Ref	Expression
rngidpropdg	|- ( ph -> ( 1r ` K ) = ( 1r ` L ) )

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

Allowed substitution hints:   V( x, y)   W( x, y)

Proof of Theorem rngidpropdg

Step	Hyp	Ref	Expression
1		rngidpropd.1	. . . 4 |- ( ph -> B = ( Base ` K ) )
2		rngidpropdg.k	. . . . 5 |- ( ph -> K e. V )
3		eqid 2177	. . . . . 6 |- (mulGrp ` K ) = (mulGrp ` K )
4		eqid 2177	. . . . . 6 |- ( Base ` K ) = ( Base ` K )
5	3, 4	mgpbasg 13141	. . . . 5 |- ( K e. V -> ( Base ` K ) = ( Base ` (mulGrp ` K ) ) )
6	2, 5	syl 14	. . . 4 |- ( ph -> ( Base ` K ) = ( Base ` (mulGrp ` K ) ) )
7	1, 6	eqtrd 2210	. . 3 |- ( ph -> B = ( Base ` (mulGrp ` K ) ) )
8		rngidpropd.2	. . . 4 |- ( ph -> B = ( Base ` L ) )
9		rngidpropdg.l	. . . . 5 |- ( ph -> L e. W )
10		eqid 2177	. . . . . 6 |- (mulGrp ` L ) = (mulGrp ` L )
11		eqid 2177	. . . . . 6 |- ( Base ` L ) = ( Base ` L )
12	10, 11	mgpbasg 13141	. . . . 5 |- ( L e. W -> ( Base ` L ) = ( Base ` (mulGrp ` L ) ) )
13	9, 12	syl 14	. . . 4 |- ( ph -> ( Base ` L ) = ( Base ` (mulGrp ` L ) ) )
14	8, 13	eqtrd 2210	. . 3 |- ( ph -> B = ( Base ` (mulGrp ` L ) ) )
15	3	mgpex 13140	. . . 4 |- ( K e. V -> (mulGrp ` K ) e. _V )
16	2, 15	syl 14	. . 3 |- ( ph -> (mulGrp ` K ) e. _V )
17	10	mgpex 13140	. . . 4 |- ( L e. W -> (mulGrp ` L ) e. _V )
18	9, 17	syl 14	. . 3 |- ( ph -> (mulGrp ` L ) e. _V )
19		rngidpropd.3	. . . 4 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )
20		eqid 2177	. . . . . . 7 |- ( .r ` K ) = ( .r ` K )
21	3, 20	mgpplusgg 13139	. . . . . 6 |- ( K e. V -> ( .r ` K ) = ( +g ` (mulGrp ` K ) ) )
22	2, 21	syl 14	. . . . 5 |- ( ph -> ( .r ` K ) = ( +g ` (mulGrp ` K ) ) )
23	22	oveqdr 5905	. . . 4 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( +g ` (mulGrp ` K ) ) y ) )
24		eqid 2177	. . . . . . 7 |- ( .r ` L ) = ( .r ` L )
25	10, 24	mgpplusgg 13139	. . . . . 6 |- ( L e. W -> ( .r ` L ) = ( +g ` (mulGrp ` L ) ) )
26	9, 25	syl 14	. . . . 5 |- ( ph -> ( .r ` L ) = ( +g ` (mulGrp ` L ) ) )
27	26	oveqdr 5905	. . . 4 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` L ) y ) = ( x ( +g ` (mulGrp ` L ) ) y ) )
28	19, 23, 27	3eqtr3d 2218	. . 3 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` (mulGrp ` K ) ) y ) = ( x ( +g ` (mulGrp ` L ) ) y ) )
29	7, 14, 16, 18, 28	grpidpropdg 12798	. 2 |- ( ph -> ( 0g ` (mulGrp ` K ) ) = ( 0g ` (mulGrp ` L ) ) )
30		eqid 2177	. . . 4 |- ( 1r ` K ) = ( 1r ` K )
31	3, 30	ringidvalg 13149	. . 3 |- ( K e. V -> ( 1r ` K ) = ( 0g ` (mulGrp ` K ) ) )
32	2, 31	syl 14	. 2 |- ( ph -> ( 1r ` K ) = ( 0g ` (mulGrp ` K ) ) )
33		eqid 2177	. . . 4 |- ( 1r ` L ) = ( 1r ` L )
34	10, 33	ringidvalg 13149	. . 3 |- ( L e. W -> ( 1r ` L ) = ( 0g ` (mulGrp ` L ) ) )
35	9, 34	syl 14	. 2 |- ( ph -> ( 1r ` L ) = ( 0g ` (mulGrp ` L ) ) )
36	29, 32, 35	3eqtr4d 2220	1 |- ( ph -> ( 1r ` K ) = ( 1r ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   _Vcvv 2739   ` cfv 5218  (class class class)co 5877   Basecbs 12464   +g cplusg 12538   .rcmulr 12539   0gc0g 12710  mulGrpcmgp 13135   1rcur 13147

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-pre-ltirr 7925  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-inn 8922  df-2 8980  df-3 8981  df-ndx 12467  df-slot 12468  df-base 12470  df-sets 12471  df-plusg 12551  df-mulr 12552  df-0g 12712  df-mgp 13136  df-ur 13148

This theorem is referenced by:  unitpropdg  13322  subrgpropd  13374  lmodprop2d  13443