Theorem rngidpropdg 14391

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 2234	. . . . . 6 |- (mulGrp ` K ) = (mulGrp ` K )
4		eqid 2234	. . . . . 6 |- ( Base ` K ) = ( Base ` K )
5	3, 4	mgpbasg 14165	. . . . 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 2267	. . 3 |- ( ph -> B = ( Base ` (mulGrp ` K ) ) )
8		rngidpropd.2	. . . 4 |- ( ph -> B = ( Base ` L ) )
9		rngidpropdg.l	. . . . 5 |- ( ph -> L e. W )
10		eqid 2234	. . . . . 6 |- (mulGrp ` L ) = (mulGrp ` L )
11		eqid 2234	. . . . . 6 |- ( Base ` L ) = ( Base ` L )
12	10, 11	mgpbasg 14165	. . . . 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 2267	. . 3 |- ( ph -> B = ( Base ` (mulGrp ` L ) ) )
15	3	mgpex 14164	. . . 4 |- ( K e. V -> (mulGrp ` K ) e. _V )
16	2, 15	syl 14	. . 3 |- ( ph -> (mulGrp ` K ) e. _V )
17	10	mgpex 14164	. . . 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 2234	. . . . . . 7 |- ( .r ` K ) = ( .r ` K )
21	3, 20	mgpplusgg 14163	. . . . . 6 |- ( K e. V -> ( .r ` K ) = ( +g ` (mulGrp ` K ) ) )
22	2, 21	syl 14	. . . . 5 |- ( ph -> ( .r ` K ) = ( +g ` (mulGrp ` K ) ) )
23	22	oveqdr 6086	. . . 4 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( +g ` (mulGrp ` K ) ) y ) )
24		eqid 2234	. . . . . . 7 |- ( .r ` L ) = ( .r ` L )
25	10, 24	mgpplusgg 14163	. . . . . 6 |- ( L e. W -> ( .r ` L ) = ( +g ` (mulGrp ` L ) ) )
26	9, 25	syl 14	. . . . 5 |- ( ph -> ( .r ` L ) = ( +g ` (mulGrp ` L ) ) )
27	26	oveqdr 6086	. . . 4 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` L ) y ) = ( x ( +g ` (mulGrp ` L ) ) y ) )
28	19, 23, 27	3eqtr3d 2275	. . 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 13637	. 2 |- ( ph -> ( 0g ` (mulGrp ` K ) ) = ( 0g ` (mulGrp ` L ) ) )
30		eqid 2234	. . . 4 |- ( 1r ` K ) = ( 1r ` K )
31	3, 30	ringidvalg 14204	. . 3 |- ( K e. V -> ( 1r ` K ) = ( 0g ` (mulGrp ` K ) ) )
32	2, 31	syl 14	. 2 |- ( ph -> ( 1r ` K ) = ( 0g ` (mulGrp ` K ) ) )
33		eqid 2234	. . . 4 |- ( 1r ` L ) = ( 1r ` L )
34	10, 33	ringidvalg 14204	. . 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 2277	1 |- ( ph -> ( 1r ` K ) = ( 1r ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   _Vcvv 2815   ` cfv 5357  (class class class)co 6058   Basecbs 13296   +g cplusg 13374   .rcmulr 13375   0gc0g 13553  mulGrpcmgp 14159   1rcur 14202

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-ima 4767  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-mgp 14160  df-ur 14203

This theorem is referenced by:  unitpropdg  14393  subrgpropd  14499  lmodprop2d  14622