Theorem invrpropdg 14153

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 2230	. . . 4 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐾))
2		eqidd 2230	. . . 4 ⊢ (𝜑 → ((mulGrp‘𝐾) ↾s (Unit‘𝐾)) = ((mulGrp‘𝐾) ↾s (Unit‘𝐾)))
3		unitpropdg.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Ring)
4		ringsrg 14050	. . . . 5 ⊢ (𝐾 ∈ Ring → 𝐾 ∈ SRing)
5	3, 4	syl 14	. . . 4 ⊢ (𝜑 → 𝐾 ∈ SRing)
6	1, 2, 5	unitgrpbasd 14119	. . 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 14152	. . . 4 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐿))
12		eqidd 2230	. . . . 5 ⊢ (𝜑 → (Unit‘𝐿) = (Unit‘𝐿))
13		eqidd 2230	. . . . 5 ⊢ (𝜑 → ((mulGrp‘𝐿) ↾s (Unit‘𝐿)) = ((mulGrp‘𝐿) ↾s (Unit‘𝐿)))
14		ringsrg 14050	. . . . . 6 ⊢ (𝐿 ∈ Ring → 𝐿 ∈ SRing)
15	10, 14	syl 14	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ SRing)
16	12, 13, 15	unitgrpbasd 14119	. . . 4 ⊢ (𝜑 → (Unit‘𝐿) = (Base‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))))
17	11, 16	eqtrd 2262	. . 3 ⊢ (𝜑 → (Unit‘𝐾) = (Base‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))))
18		eqid 2229	. . . . . 6 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
19	18	ringmgp 14005	. . . . 5 ⊢ (𝐾 ∈ Ring → (mulGrp‘𝐾) ∈ Mnd)
20	3, 19	syl 14	. . . 4 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ Mnd)
21		basfn 13131	. . . . . . 7 ⊢ Base Fn V
22	3	elexd 2814	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ V)
23		funfvex 5652	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝐾 ∈ dom Base) → (Base‘𝐾) ∈ V)
24	23	funfni 5429	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝐾 ∈ V) → (Base‘𝐾) ∈ V)
25	21, 22, 24	sylancr 414	. . . . . 6 ⊢ (𝜑 → (Base‘𝐾) ∈ V)
26	7, 25	eqeltrd 2306	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
27	7, 1, 5	unitssd 14113	. . . . 5 ⊢ (𝜑 → (Unit‘𝐾) ⊆ 𝐵)
28	26, 27	ssexd 4227	. . . 4 ⊢ (𝜑 → (Unit‘𝐾) ∈ V)
29		ressex 13138	. . . 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 2229	. . . . . 6 ⊢ (mulGrp‘𝐿) = (mulGrp‘𝐿)
32	31	ringmgp 14005	. . . . 5 ⊢ (𝐿 ∈ Ring → (mulGrp‘𝐿) ∈ Mnd)
33	10, 32	syl 14	. . . 4 ⊢ (𝜑 → (mulGrp‘𝐿) ∈ Mnd)
34	11, 28	eqeltrrd 2307	. . . 4 ⊢ (𝜑 → (Unit‘𝐿) ∈ V)
35		ressex 13138	. . . 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 3225	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Unit‘𝐾)) → 𝑥 ∈ 𝐵)
38	27	sselda 3225	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (Unit‘𝐾)) → 𝑦 ∈ 𝐵)
39	37, 38	anim12dan 602	. . . . 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 2229	. . . . . . . 8 ⊢ (.r‘𝐾) = (.r‘𝐾)
42	18, 41	mgpplusgg 13927	. . . . . . 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 13202	. . . . 5 ⊢ (𝜑 → (.r‘𝐾) = (+g‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))))
45	44	oveqdr 6041	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Unit‘𝐾) ∧ 𝑦 ∈ (Unit‘𝐾))) → (𝑥(.r‘𝐾)𝑦) = (𝑥(+g‘((mulGrp‘𝐾) ↾s (Unit‘𝐾)))𝑦))
46		eqid 2229	. . . . . . . 8 ⊢ (.r‘𝐿) = (.r‘𝐿)
47	31, 46	mgpplusgg 13927	. . . . . . 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 13202	. . . . 5 ⊢ (𝜑 → (.r‘𝐿) = (+g‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))))
50	49	oveqdr 6041	. . . 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 2270	. . 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 13648	. 2 ⊢ (𝜑 → (invg‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))) = (invg‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))))
53		eqidd 2230	. . 3 ⊢ (𝜑 → (invr‘𝐾) = (invr‘𝐾))
54	1, 2, 53, 3	invrfvald 14126	. 2 ⊢ (𝜑 → (invr‘𝐾) = (invg‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))))
55		eqidd 2230	. . 3 ⊢ (𝜑 → (invr‘𝐿) = (invr‘𝐿))
56	12, 13, 55, 10	invrfvald 14126	. 2 ⊢ (𝜑 → (invr‘𝐿) = (invg‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))))
57	52, 54, 56	3eqtr4d 2272	1 ⊢ (𝜑 → (invr‘𝐾) = (invr‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  Vcvv 2800   Fn wfn 5319  ‘cfv 5324  (class class class)co 6013  Basecbs 13072   ↾_s cress 13073  +_gcplusg 13150  ._rcmulr 13151  Mndcmnd 13489  inv_gcminusg 13574  mulGrpcmgp 13923  SRingcsrg 13966  Ringcrg 13999  Unitcui 14090  inv_rcinvr 14124

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-pre-ltirr 8134  ax-pre-lttrn 8136  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-tpos 6406  df-pnf 8206  df-mnf 8207  df-ltxr 8209  df-inn 9134  df-2 9192  df-3 9193  df-ndx 13075  df-slot 13076  df-base 13078  df-sets 13079  df-iress 13080  df-plusg 13163  df-mulr 13164  df-0g 13331  df-mgm 13429  df-sgrp 13475  df-mnd 13490  df-grp 13576  df-minusg 13577  df-cmn 13863  df-abl 13864  df-mgp 13924  df-ur 13963  df-srg 13967  df-ring 14001  df-oppr 14071  df-dvdsr 14092  df-unit 14093  df-invr 14125

This theorem is referenced by: (None)