Theorem rngidpropdg 14453

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 2238	. . . . . 6 |- (mulGrp ` K ) = (mulGrp ` K )
4		eqid 2238	. . . . . 6 |- ( Base ` K ) = ( Base ` K )
5	3, 4	mgpbasg 14224	. . . . 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 2271	. . 3 |- ( ph -> B = ( Base ` (mulGrp ` K ) ) )
8		rngidpropd.2	. . . 4 |- ( ph -> B = ( Base ` L ) )
9		rngidpropdg.l	. . . . 5 |- ( ph -> L e. W )
10		eqid 2238	. . . . . 6 |- (mulGrp ` L ) = (mulGrp ` L )
11		eqid 2238	. . . . . 6 |- ( Base ` L ) = ( Base ` L )
12	10, 11	mgpbasg 14224	. . . . 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 2271	. . 3 |- ( ph -> B = ( Base ` (mulGrp ` L ) ) )
15	3	mgpex 14223	. . . 4 |- ( K e. V -> (mulGrp ` K ) e. _V )
16	2, 15	syl 14	. . 3 |- ( ph -> (mulGrp ` K ) e. _V )
17	10	mgpex 14223	. . . 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 2238	. . . . . . 7 |- ( .r ` K ) = ( .r ` K )
21	3, 20	mgpplusgg 14221	. . . . . 6 |- ( K e. V -> ( .r ` K ) = ( +g ` (mulGrp ` K ) ) )
22	2, 21	syl 14	. . . . 5 |- ( ph -> ( .r ` K ) = ( +g ` (mulGrp ` K ) ) )
23	22	oveqdr 6113	. . . 4 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( +g ` (mulGrp ` K ) ) y ) )
24		eqid 2238	. . . . . . 7 |- ( .r ` L ) = ( .r ` L )
25	10, 24	mgpplusgg 14221	. . . . . 6 |- ( L e. W -> ( .r ` L ) = ( +g ` (mulGrp ` L ) ) )
26	9, 25	syl 14	. . . . 5 |- ( ph -> ( .r ` L ) = ( +g ` (mulGrp ` L ) ) )
27	26	oveqdr 6113	. . . 4 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` L ) y ) = ( x ( +g ` (mulGrp ` L ) ) y ) )
28	19, 23, 27	3eqtr3d 2279	. . 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 13694	. 2 |- ( ph -> ( 0g ` (mulGrp ` K ) ) = ( 0g ` (mulGrp ` L ) ) )
30		eqid 2238	. . . 4 |- ( 1r ` K ) = ( 1r ` K )
31	3, 30	ringidvalg 14264	. . 3 |- ( K e. V -> ( 1r ` K ) = ( 0g ` (mulGrp ` K ) ) )
32	2, 31	syl 14	. 2 |- ( ph -> ( 1r ` K ) = ( 0g ` (mulGrp ` K ) ) )
33		eqid 2238	. . . 4 |- ( 1r ` L ) = ( 1r ` L )
34	10, 33	ringidvalg 14264	. . 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 2281	1 |- ( ph -> ( 1r ` K ) = ( 1r ` L ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   ` cfv 5377  (class class class)co 6085   Basecbs 13352   +g cplusg 13431   .rcmulr 13432   0gc0g 13610  mulGrpcmgp 14217   1rcur 14262

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9305  df-2 9363  df-3 9364  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-plusg 13444  df-mulr 13445  df-0g 13612  df-mgp 14218  df-ur 14263

This theorem is used by:  unitpropdg  14455  subrgpropd  14561  lmodprop2d  14685