Theorem invrpropdg 13645

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 2194	. . . 4 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐾))
2		eqidd 2194	. . . 4 ⊢ (𝜑 → ((mulGrp‘𝐾) ↾s (Unit‘𝐾)) = ((mulGrp‘𝐾) ↾s (Unit‘𝐾)))
3		unitpropdg.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Ring)
4		ringsrg 13543	. . . . 5 ⊢ (𝐾 ∈ Ring → 𝐾 ∈ SRing)
5	3, 4	syl 14	. . . 4 ⊢ (𝜑 → 𝐾 ∈ SRing)
6	1, 2, 5	unitgrpbasd 13611	. . 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 13644	. . . 4 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐿))
12		eqidd 2194	. . . . 5 ⊢ (𝜑 → (Unit‘𝐿) = (Unit‘𝐿))
13		eqidd 2194	. . . . 5 ⊢ (𝜑 → ((mulGrp‘𝐿) ↾s (Unit‘𝐿)) = ((mulGrp‘𝐿) ↾s (Unit‘𝐿)))
14		ringsrg 13543	. . . . . 6 ⊢ (𝐿 ∈ Ring → 𝐿 ∈ SRing)
15	10, 14	syl 14	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ SRing)
16	12, 13, 15	unitgrpbasd 13611	. . . 4 ⊢ (𝜑 → (Unit‘𝐿) = (Base‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))))
17	11, 16	eqtrd 2226	. . 3 ⊢ (𝜑 → (Unit‘𝐾) = (Base‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))))
18		eqid 2193	. . . . . 6 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
19	18	ringmgp 13498	. . . . 5 ⊢ (𝐾 ∈ Ring → (mulGrp‘𝐾) ∈ Mnd)
20	3, 19	syl 14	. . . 4 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ Mnd)
21		basfn 12676	. . . . . . 7 ⊢ Base Fn V
22	3	elexd 2773	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ V)
23		funfvex 5571	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝐾 ∈ dom Base) → (Base‘𝐾) ∈ V)
24	23	funfni 5354	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝐾 ∈ V) → (Base‘𝐾) ∈ V)
25	21, 22, 24	sylancr 414	. . . . . 6 ⊢ (𝜑 → (Base‘𝐾) ∈ V)
26	7, 25	eqeltrd 2270	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
27	7, 1, 5	unitssd 13605	. . . . 5 ⊢ (𝜑 → (Unit‘𝐾) ⊆ 𝐵)
28	26, 27	ssexd 4169	. . . 4 ⊢ (𝜑 → (Unit‘𝐾) ∈ V)
29		ressex 12683	. . . 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 2193	. . . . . 6 ⊢ (mulGrp‘𝐿) = (mulGrp‘𝐿)
32	31	ringmgp 13498	. . . . 5 ⊢ (𝐿 ∈ Ring → (mulGrp‘𝐿) ∈ Mnd)
33	10, 32	syl 14	. . . 4 ⊢ (𝜑 → (mulGrp‘𝐿) ∈ Mnd)
34	11, 28	eqeltrrd 2271	. . . 4 ⊢ (𝜑 → (Unit‘𝐿) ∈ V)
35		ressex 12683	. . . 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 3179	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Unit‘𝐾)) → 𝑥 ∈ 𝐵)
38	27	sselda 3179	. . . . . 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 2193	. . . . . . . 8 ⊢ (.r‘𝐾) = (.r‘𝐾)
42	18, 41	mgpplusgg 13420	. . . . . . 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 12746	. . . . 5 ⊢ (𝜑 → (.r‘𝐾) = (+g‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))))
45	44	oveqdr 5946	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Unit‘𝐾) ∧ 𝑦 ∈ (Unit‘𝐾))) → (𝑥(.r‘𝐾)𝑦) = (𝑥(+g‘((mulGrp‘𝐾) ↾s (Unit‘𝐾)))𝑦))
46		eqid 2193	. . . . . . . 8 ⊢ (.r‘𝐿) = (.r‘𝐿)
47	31, 46	mgpplusgg 13420	. . . . . . 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 12746	. . . . 5 ⊢ (𝜑 → (.r‘𝐿) = (+g‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))))
50	49	oveqdr 5946	. . . 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 2234	. . 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 13147	. 2 ⊢ (𝜑 → (invg‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))) = (invg‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))))
53		eqidd 2194	. . 3 ⊢ (𝜑 → (invr‘𝐾) = (invr‘𝐾))
54	1, 2, 53, 3	invrfvald 13618	. 2 ⊢ (𝜑 → (invr‘𝐾) = (invg‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))))
55		eqidd 2194	. . 3 ⊢ (𝜑 → (invr‘𝐿) = (invr‘𝐿))
56	12, 13, 55, 10	invrfvald 13618	. 2 ⊢ (𝜑 → (invr‘𝐿) = (invg‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))))
57	52, 54, 56	3eqtr4d 2236	1 ⊢ (𝜑 → (invr‘𝐾) = (invr‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  Vcvv 2760   Fn wfn 5249  ‘cfv 5254  (class class class)co 5918  Basecbs 12618   ↾_s cress 12619  +_gcplusg 12695  ._rcmulr 12696  Mndcmnd 12997  inv_gcminusg 13073  mulGrpcmgp 13416  SRingcsrg 13459  Ringcrg 13492  Unitcui 13583  inv_rcinvr 13616

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-tpos 6298  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-iress 12626  df-plusg 12708  df-mulr 12709  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076  df-cmn 13356  df-abl 13357  df-mgp 13417  df-ur 13456  df-srg 13460  df-ring 13494  df-oppr 13564  df-dvdsr 13585  df-unit 13586  df-invr 13617

This theorem is referenced by: (None)