Theorem invrpropdg 13316

Description: The ring inverse function depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.) (Revised by Mario Carneiro, 5-Oct-2015.)

Hypotheses

Ref	Expression
unitpropdg.1	⊢ (𝜑 → 𝐵 = (Base‘𝐾))
unitpropdg.2	⊢ (𝜑 → 𝐵 = (Base‘𝐿))
unitpropdg.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))
unitpropdg.k	⊢ (𝜑 → 𝐾 ∈ Ring)
unitpropdg.l	⊢ (𝜑 → 𝐿 ∈ Ring)

Assertion

Ref	Expression
invrpropdg	⊢ (𝜑 → (invr‘𝐾) = (invr‘𝐿))

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

Proof of Theorem invrpropdg

Step	Hyp	Ref	Expression
1		eqidd 2178	. . . 4 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐾))
2		eqidd 2178	. . . 4 ⊢ (𝜑 → ((mulGrp‘𝐾) ↾s (Unit‘𝐾)) = ((mulGrp‘𝐾) ↾s (Unit‘𝐾)))
3		unitpropdg.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Ring)
4		ringsrg 13222	. . . . 5 ⊢ (𝐾 ∈ Ring → 𝐾 ∈ SRing)
5	3, 4	syl 14	. . . 4 ⊢ (𝜑 → 𝐾 ∈ SRing)
6	1, 2, 5	unitgrpbasd 13282	. . 3 ⊢ (𝜑 → (Unit‘𝐾) = (Base‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))))
7		unitpropdg.1	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
8		unitpropdg.2	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
9		unitpropdg.3	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))
10		unitpropdg.l	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ Ring)
11	7, 8, 9, 3, 10	unitpropdg 13315	. . . 4 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐿))
12		eqidd 2178	. . . . 5 ⊢ (𝜑 → (Unit‘𝐿) = (Unit‘𝐿))
13		eqidd 2178	. . . . 5 ⊢ (𝜑 → ((mulGrp‘𝐿) ↾s (Unit‘𝐿)) = ((mulGrp‘𝐿) ↾s (Unit‘𝐿)))
14		ringsrg 13222	. . . . . 6 ⊢ (𝐿 ∈ Ring → 𝐿 ∈ SRing)
15	10, 14	syl 14	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ SRing)
16	12, 13, 15	unitgrpbasd 13282	. . . 4 ⊢ (𝜑 → (Unit‘𝐿) = (Base‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))))
17	11, 16	eqtrd 2210	. . 3 ⊢ (𝜑 → (Unit‘𝐾) = (Base‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))))
18		eqid 2177	. . . . . 6 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
19	18	ringmgp 13183	. . . . 5 ⊢ (𝐾 ∈ Ring → (mulGrp‘𝐾) ∈ Mnd)
20	3, 19	syl 14	. . . 4 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ Mnd)
21		basfn 12519	. . . . . . 7 ⊢ Base Fn V
22	3	elexd 2750	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ V)
23		funfvex 5532	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝐾 ∈ dom Base) → (Base‘𝐾) ∈ V)
24	23	funfni 5316	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝐾 ∈ V) → (Base‘𝐾) ∈ V)
25	21, 22, 24	sylancr 414	. . . . . 6 ⊢ (𝜑 → (Base‘𝐾) ∈ V)
26	7, 25	eqeltrd 2254	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
27	7, 1, 5	unitssd 13276	. . . . 5 ⊢ (𝜑 → (Unit‘𝐾) ⊆ 𝐵)
28	26, 27	ssexd 4143	. . . 4 ⊢ (𝜑 → (Unit‘𝐾) ∈ V)
29		ressex 12524	. . . 4 ⊢ (((mulGrp‘𝐾) ∈ Mnd ∧ (Unit‘𝐾) ∈ V) → ((mulGrp‘𝐾) ↾s (Unit‘𝐾)) ∈ V)
30	20, 28, 29	syl2anc 411	. . 3 ⊢ (𝜑 → ((mulGrp‘𝐾) ↾s (Unit‘𝐾)) ∈ V)
31		eqid 2177	. . . . . 6 ⊢ (mulGrp‘𝐿) = (mulGrp‘𝐿)
32	31	ringmgp 13183	. . . . 5 ⊢ (𝐿 ∈ Ring → (mulGrp‘𝐿) ∈ Mnd)
33	10, 32	syl 14	. . . 4 ⊢ (𝜑 → (mulGrp‘𝐿) ∈ Mnd)
34	11, 28	eqeltrrd 2255	. . . 4 ⊢ (𝜑 → (Unit‘𝐿) ∈ V)
35		ressex 12524	. . . 4 ⊢ (((mulGrp‘𝐿) ∈ Mnd ∧ (Unit‘𝐿) ∈ V) → ((mulGrp‘𝐿) ↾s (Unit‘𝐿)) ∈ V)
36	33, 34, 35	syl2anc 411	. . 3 ⊢ (𝜑 → ((mulGrp‘𝐿) ↾s (Unit‘𝐿)) ∈ V)
37	27	sselda 3155	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Unit‘𝐾)) → 𝑥 ∈ 𝐵)
38	27	sselda 3155	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (Unit‘𝐾)) → 𝑦 ∈ 𝐵)
39	37, 38	anim12dan 600	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Unit‘𝐾) ∧ 𝑦 ∈ (Unit‘𝐾))) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
40	39, 9	syldan 282	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Unit‘𝐾) ∧ 𝑦 ∈ (Unit‘𝐾))) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))
41		eqid 2177	. . . . . . . 8 ⊢ (.r‘𝐾) = (.r‘𝐾)
42	18, 41	mgpplusgg 13132	. . . . . . 7 ⊢ (𝐾 ∈ Ring → (.r‘𝐾) = (+g‘(mulGrp‘𝐾)))
43	3, 42	syl 14	. . . . . 6 ⊢ (𝜑 → (.r‘𝐾) = (+g‘(mulGrp‘𝐾)))
44	2, 43, 28, 20	ressplusgd 12586	. . . . 5 ⊢ (𝜑 → (.r‘𝐾) = (+g‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))))
45	44	oveqdr 5902	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Unit‘𝐾) ∧ 𝑦 ∈ (Unit‘𝐾))) → (𝑥(.r‘𝐾)𝑦) = (𝑥(+g‘((mulGrp‘𝐾) ↾s (Unit‘𝐾)))𝑦))
46		eqid 2177	. . . . . . . 8 ⊢ (.r‘𝐿) = (.r‘𝐿)
47	31, 46	mgpplusgg 13132	. . . . . . 7 ⊢ (𝐿 ∈ Ring → (.r‘𝐿) = (+g‘(mulGrp‘𝐿)))
48	10, 47	syl 14	. . . . . 6 ⊢ (𝜑 → (.r‘𝐿) = (+g‘(mulGrp‘𝐿)))
49	13, 48, 34, 33	ressplusgd 12586	. . . . 5 ⊢ (𝜑 → (.r‘𝐿) = (+g‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))))
50	49	oveqdr 5902	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Unit‘𝐾) ∧ 𝑦 ∈ (Unit‘𝐾))) → (𝑥(.r‘𝐿)𝑦) = (𝑥(+g‘((mulGrp‘𝐿) ↾s (Unit‘𝐿)))𝑦))
51	40, 45, 50	3eqtr3d 2218	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Unit‘𝐾) ∧ 𝑦 ∈ (Unit‘𝐾))) → (𝑥(+g‘((mulGrp‘𝐾) ↾s (Unit‘𝐾)))𝑦) = (𝑥(+g‘((mulGrp‘𝐿) ↾s (Unit‘𝐿)))𝑦))
52	6, 17, 30, 36, 51	grpinvpropdg 12944	. 2 ⊢ (𝜑 → (invg‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))) = (invg‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))))
53		eqidd 2178	. . 3 ⊢ (𝜑 → (invr‘𝐾) = (invr‘𝐾))
54	1, 2, 53, 3	invrfvald 13289	. 2 ⊢ (𝜑 → (invr‘𝐾) = (invg‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))))
55		eqidd 2178	. . 3 ⊢ (𝜑 → (invr‘𝐿) = (invr‘𝐿))
56	12, 13, 55, 10	invrfvald 13289	. 2 ⊢ (𝜑 → (invr‘𝐿) = (invg‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))))
57	52, 54, 56	3eqtr4d 2220	1 ⊢ (𝜑 → (invr‘𝐾) = (invr‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148  Vcvv 2737   Fn wfn 5211  ‘cfv 5216  (class class class)co 5874  Basecbs 12461   ↾_s cress 12462  +_gcplusg 12535  ._rcmulr 12536  Mndcmnd 12816  inv_gcminusg 12877  mulGrpcmgp 13128  SRingcsrg 13144  Ringcrg 13177  Unitcui 13254  inv_rcinvr 13287

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-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-pre-ltirr 7922  ax-pre-lttrn 7924  ax-pre-ltadd 7926

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-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-tpos 6245  df-pnf 7993  df-mnf 7994  df-ltxr 7996  df-inn 8919  df-2 8977  df-3 8978  df-ndx 12464  df-slot 12465  df-base 12467  df-sets 12468  df-iress 12469  df-plusg 12548  df-mulr 12549  df-0g 12706  df-mgm 12774  df-sgrp 12807  df-mnd 12817  df-grp 12879  df-minusg 12880  df-cmn 13088  df-abl 13089  df-mgp 13129  df-ur 13141  df-srg 13145  df-ring 13179  df-oppr 13238  df-dvdsr 13256  df-unit 13257  df-invr 13288

This theorem is referenced by: (None)