Theorem invrpropdg 13705

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 2197	. . . 4 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐾))
2		eqidd 2197	. . . 4 ⊢ (𝜑 → ((mulGrp‘𝐾) ↾s (Unit‘𝐾)) = ((mulGrp‘𝐾) ↾s (Unit‘𝐾)))
3		unitpropdg.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Ring)
4		ringsrg 13603	. . . . 5 ⊢ (𝐾 ∈ Ring → 𝐾 ∈ SRing)
5	3, 4	syl 14	. . . 4 ⊢ (𝜑 → 𝐾 ∈ SRing)
6	1, 2, 5	unitgrpbasd 13671	. . 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 13704	. . . 4 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐿))
12		eqidd 2197	. . . . 5 ⊢ (𝜑 → (Unit‘𝐿) = (Unit‘𝐿))
13		eqidd 2197	. . . . 5 ⊢ (𝜑 → ((mulGrp‘𝐿) ↾s (Unit‘𝐿)) = ((mulGrp‘𝐿) ↾s (Unit‘𝐿)))
14		ringsrg 13603	. . . . . 6 ⊢ (𝐿 ∈ Ring → 𝐿 ∈ SRing)
15	10, 14	syl 14	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ SRing)
16	12, 13, 15	unitgrpbasd 13671	. . . 4 ⊢ (𝜑 → (Unit‘𝐿) = (Base‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))))
17	11, 16	eqtrd 2229	. . 3 ⊢ (𝜑 → (Unit‘𝐾) = (Base‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))))
18		eqid 2196	. . . . . 6 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
19	18	ringmgp 13558	. . . . 5 ⊢ (𝐾 ∈ Ring → (mulGrp‘𝐾) ∈ Mnd)
20	3, 19	syl 14	. . . 4 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ Mnd)
21		basfn 12736	. . . . . . 7 ⊢ Base Fn V
22	3	elexd 2776	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ V)
23		funfvex 5575	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝐾 ∈ dom Base) → (Base‘𝐾) ∈ V)
24	23	funfni 5358	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝐾 ∈ V) → (Base‘𝐾) ∈ V)
25	21, 22, 24	sylancr 414	. . . . . 6 ⊢ (𝜑 → (Base‘𝐾) ∈ V)
26	7, 25	eqeltrd 2273	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
27	7, 1, 5	unitssd 13665	. . . . 5 ⊢ (𝜑 → (Unit‘𝐾) ⊆ 𝐵)
28	26, 27	ssexd 4173	. . . 4 ⊢ (𝜑 → (Unit‘𝐾) ∈ V)
29		ressex 12743	. . . 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 2196	. . . . . 6 ⊢ (mulGrp‘𝐿) = (mulGrp‘𝐿)
32	31	ringmgp 13558	. . . . 5 ⊢ (𝐿 ∈ Ring → (mulGrp‘𝐿) ∈ Mnd)
33	10, 32	syl 14	. . . 4 ⊢ (𝜑 → (mulGrp‘𝐿) ∈ Mnd)
34	11, 28	eqeltrrd 2274	. . . 4 ⊢ (𝜑 → (Unit‘𝐿) ∈ V)
35		ressex 12743	. . . 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 3183	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Unit‘𝐾)) → 𝑥 ∈ 𝐵)
38	27	sselda 3183	. . . . . 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 2196	. . . . . . . 8 ⊢ (.r‘𝐾) = (.r‘𝐾)
42	18, 41	mgpplusgg 13480	. . . . . . 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 12806	. . . . 5 ⊢ (𝜑 → (.r‘𝐾) = (+g‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))))
45	44	oveqdr 5950	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Unit‘𝐾) ∧ 𝑦 ∈ (Unit‘𝐾))) → (𝑥(.r‘𝐾)𝑦) = (𝑥(+g‘((mulGrp‘𝐾) ↾s (Unit‘𝐾)))𝑦))
46		eqid 2196	. . . . . . . 8 ⊢ (.r‘𝐿) = (.r‘𝐿)
47	31, 46	mgpplusgg 13480	. . . . . . 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 12806	. . . . 5 ⊢ (𝜑 → (.r‘𝐿) = (+g‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))))
50	49	oveqdr 5950	. . . 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 2237	. . 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 13207	. 2 ⊢ (𝜑 → (invg‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))) = (invg‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))))
53		eqidd 2197	. . 3 ⊢ (𝜑 → (invr‘𝐾) = (invr‘𝐾))
54	1, 2, 53, 3	invrfvald 13678	. 2 ⊢ (𝜑 → (invr‘𝐾) = (invg‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))))
55		eqidd 2197	. . 3 ⊢ (𝜑 → (invr‘𝐿) = (invr‘𝐿))
56	12, 13, 55, 10	invrfvald 13678	. 2 ⊢ (𝜑 → (invr‘𝐿) = (invg‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))))
57	52, 54, 56	3eqtr4d 2239	1 ⊢ (𝜑 → (invr‘𝐾) = (invr‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  Vcvv 2763   Fn wfn 5253  ‘cfv 5258  (class class class)co 5922  Basecbs 12678   ↾_s cress 12679  +_gcplusg 12755  ._rcmulr 12756  Mndcmnd 13057  inv_gcminusg 13133  mulGrpcmgp 13476  SRingcsrg 13519  Ringcrg 13552  Unitcui 13643  inv_rcinvr 13676

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-tpos 6303  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-mulr 12769  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-cmn 13416  df-abl 13417  df-mgp 13477  df-ur 13516  df-srg 13520  df-ring 13554  df-oppr 13624  df-dvdsr 13645  df-unit 13646  df-invr 13677

This theorem is referenced by: (None)