Theorem rhmpropd 19573

Description: Ring homomorphism depends only on the ring attributes of structures. (Contributed by Mario Carneiro, 12-Jun-2015.)

Hypotheses

Ref	Expression
rhmpropd.a	⊢ (𝜑 → 𝐵 = (Base‘𝐽))
rhmpropd.b	⊢ (𝜑 → 𝐶 = (Base‘𝐾))
rhmpropd.c	⊢ (𝜑 → 𝐵 = (Base‘𝐿))
rhmpropd.d	⊢ (𝜑 → 𝐶 = (Base‘𝑀))
rhmpropd.e	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐽)𝑦) = (𝑥(+g‘𝐿)𝑦))
rhmpropd.f	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦))
rhmpropd.g	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐽)𝑦) = (𝑥(.r‘𝐿)𝑦))
rhmpropd.h	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝑀)𝑦))

Assertion

Ref	Expression
rhmpropd	⊢ (𝜑 → (𝐽 RingHom 𝐾) = (𝐿 RingHom 𝑀))

Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝑥,𝑀,𝑦   𝜑,𝑥,𝑦   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem rhmpropd

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rhmpropd.a	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐽))
2		rhmpropd.c	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
3		rhmpropd.e	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐽)𝑦) = (𝑥(+g‘𝐿)𝑦))
4		rhmpropd.g	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐽)𝑦) = (𝑥(.r‘𝐿)𝑦))
5	1, 2, 3, 4	ringpropd 19334	. . . . 5 ⊢ (𝜑 → (𝐽 ∈ Ring ↔ 𝐿 ∈ Ring))
6		rhmpropd.b	. . . . . 6 ⊢ (𝜑 → 𝐶 = (Base‘𝐾))
7		rhmpropd.d	. . . . . 6 ⊢ (𝜑 → 𝐶 = (Base‘𝑀))
8		rhmpropd.f	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦))
9		rhmpropd.h	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝑀)𝑦))
10	6, 7, 8, 9	ringpropd 19334	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ Ring ↔ 𝑀 ∈ Ring))
11	5, 10	anbi12d 632	. . . 4 ⊢ (𝜑 → ((𝐽 ∈ Ring ∧ 𝐾 ∈ Ring) ↔ (𝐿 ∈ Ring ∧ 𝑀 ∈ Ring)))
12	1, 6, 2, 7, 3, 8	ghmpropd 18398	. . . . . 6 ⊢ (𝜑 → (𝐽 GrpHom 𝐾) = (𝐿 GrpHom 𝑀))
13	12	eleq2d 2900	. . . . 5 ⊢ (𝜑 → (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ 𝑓 ∈ (𝐿 GrpHom 𝑀)))
14		eqid 2823	. . . . . . . . 9 ⊢ (mulGrp‘𝐽) = (mulGrp‘𝐽)
15		eqid 2823	. . . . . . . . 9 ⊢ (Base‘𝐽) = (Base‘𝐽)
16	14, 15	mgpbas 19247	. . . . . . . 8 ⊢ (Base‘𝐽) = (Base‘(mulGrp‘𝐽))
17	1, 16	syl6eq 2874	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘(mulGrp‘𝐽)))
18		eqid 2823	. . . . . . . . 9 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
19		eqid 2823	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
20	18, 19	mgpbas 19247	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
21	6, 20	syl6eq 2874	. . . . . . 7 ⊢ (𝜑 → 𝐶 = (Base‘(mulGrp‘𝐾)))
22		eqid 2823	. . . . . . . . 9 ⊢ (mulGrp‘𝐿) = (mulGrp‘𝐿)
23		eqid 2823	. . . . . . . . 9 ⊢ (Base‘𝐿) = (Base‘𝐿)
24	22, 23	mgpbas 19247	. . . . . . . 8 ⊢ (Base‘𝐿) = (Base‘(mulGrp‘𝐿))
25	2, 24	syl6eq 2874	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘(mulGrp‘𝐿)))
26		eqid 2823	. . . . . . . . 9 ⊢ (mulGrp‘𝑀) = (mulGrp‘𝑀)
27		eqid 2823	. . . . . . . . 9 ⊢ (Base‘𝑀) = (Base‘𝑀)
28	26, 27	mgpbas 19247	. . . . . . . 8 ⊢ (Base‘𝑀) = (Base‘(mulGrp‘𝑀))
29	7, 28	syl6eq 2874	. . . . . . 7 ⊢ (𝜑 → 𝐶 = (Base‘(mulGrp‘𝑀)))
30		eqid 2823	. . . . . . . . . 10 ⊢ (.r‘𝐽) = (.r‘𝐽)
31	14, 30	mgpplusg 19245	. . . . . . . . 9 ⊢ (.r‘𝐽) = (+g‘(mulGrp‘𝐽))
32	31	oveqi 7171	. . . . . . . 8 ⊢ (𝑥(.r‘𝐽)𝑦) = (𝑥(+g‘(mulGrp‘𝐽))𝑦)
33		eqid 2823	. . . . . . . . . 10 ⊢ (.r‘𝐿) = (.r‘𝐿)
34	22, 33	mgpplusg 19245	. . . . . . . . 9 ⊢ (.r‘𝐿) = (+g‘(mulGrp‘𝐿))
35	34	oveqi 7171	. . . . . . . 8 ⊢ (𝑥(.r‘𝐿)𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦)
36	4, 32, 35	3eqtr3g 2881	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘(mulGrp‘𝐽))𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦))
37		eqid 2823	. . . . . . . . . 10 ⊢ (.r‘𝐾) = (.r‘𝐾)
38	18, 37	mgpplusg 19245	. . . . . . . . 9 ⊢ (.r‘𝐾) = (+g‘(mulGrp‘𝐾))
39	38	oveqi 7171	. . . . . . . 8 ⊢ (𝑥(.r‘𝐾)𝑦) = (𝑥(+g‘(mulGrp‘𝐾))𝑦)
40		eqid 2823	. . . . . . . . . 10 ⊢ (.r‘𝑀) = (.r‘𝑀)
41	26, 40	mgpplusg 19245	. . . . . . . . 9 ⊢ (.r‘𝑀) = (+g‘(mulGrp‘𝑀))
42	41	oveqi 7171	. . . . . . . 8 ⊢ (𝑥(.r‘𝑀)𝑦) = (𝑥(+g‘(mulGrp‘𝑀))𝑦)
43	9, 39, 42	3eqtr3g 2881	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+g‘(mulGrp‘𝐾))𝑦) = (𝑥(+g‘(mulGrp‘𝑀))𝑦))
44	17, 21, 25, 29, 36, 43	mhmpropd 17964	. . . . . 6 ⊢ (𝜑 → ((mulGrp‘𝐽) MndHom (mulGrp‘𝐾)) = ((mulGrp‘𝐿) MndHom (mulGrp‘𝑀)))
45	44	eleq2d 2900	. . . . 5 ⊢ (𝜑 → (𝑓 ∈ ((mulGrp‘𝐽) MndHom (mulGrp‘𝐾)) ↔ 𝑓 ∈ ((mulGrp‘𝐿) MndHom (mulGrp‘𝑀))))
46	13, 45	anbi12d 632	. . . 4 ⊢ (𝜑 → ((𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝑓 ∈ ((mulGrp‘𝐽) MndHom (mulGrp‘𝐾))) ↔ (𝑓 ∈ (𝐿 GrpHom 𝑀) ∧ 𝑓 ∈ ((mulGrp‘𝐿) MndHom (mulGrp‘𝑀)))))
47	11, 46	anbi12d 632	. . 3 ⊢ (𝜑 → (((𝐽 ∈ Ring ∧ 𝐾 ∈ Ring) ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝑓 ∈ ((mulGrp‘𝐽) MndHom (mulGrp‘𝐾)))) ↔ ((𝐿 ∈ Ring ∧ 𝑀 ∈ Ring) ∧ (𝑓 ∈ (𝐿 GrpHom 𝑀) ∧ 𝑓 ∈ ((mulGrp‘𝐿) MndHom (mulGrp‘𝑀))))))
48	14, 18	isrhm 19475	. . 3 ⊢ (𝑓 ∈ (𝐽 RingHom 𝐾) ↔ ((𝐽 ∈ Ring ∧ 𝐾 ∈ Ring) ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝑓 ∈ ((mulGrp‘𝐽) MndHom (mulGrp‘𝐾)))))
49	22, 26	isrhm 19475	. . 3 ⊢ (𝑓 ∈ (𝐿 RingHom 𝑀) ↔ ((𝐿 ∈ Ring ∧ 𝑀 ∈ Ring) ∧ (𝑓 ∈ (𝐿 GrpHom 𝑀) ∧ 𝑓 ∈ ((mulGrp‘𝐿) MndHom (mulGrp‘𝑀)))))
50	47, 48, 49	3bitr4g 316	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝐽 RingHom 𝐾) ↔ 𝑓 ∈ (𝐿 RingHom 𝑀)))
51	50	eqrdv 2821	1 ⊢ (𝜑 → (𝐽 RingHom 𝐾) = (𝐿 RingHom 𝑀))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ‘cfv 6357  (class class class)co 7158  Basecbs 16485  +_gcplusg 16567  ._rcmulr 16568   MndHom cmhm 17956   GrpHom cghm 18357  mulGrpcmgp 19241  Ringcrg 19299   RingHom crh 19466

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-er 8291  df-map 8410  df-en 8512  df-dom 8513  df-sdom 8514  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-nn 11641  df-2 11703  df-ndx 16488  df-slot 16489  df-base 16491  df-sets 16492  df-plusg 16580  df-0g 16717  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-mhm 17958  df-grp 18108  df-ghm 18358  df-mgp 19242  df-ur 19254  df-ring 19301  df-rnghom 19469

This theorem is referenced by:  evls1rhm  20487  evl1rhm  20497  zrhpropd  20664