Theorem invrpropdg 14227

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 2232	. . . 4 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐾))
2		eqidd 2232	. . . 4 ⊢ (𝜑 → ((mulGrp‘𝐾) ↾s (Unit‘𝐾)) = ((mulGrp‘𝐾) ↾s (Unit‘𝐾)))
3		unitpropdg.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Ring)
4		ringsrg 14124	. . . . 5 ⊢ (𝐾 ∈ Ring → 𝐾 ∈ SRing)
5	3, 4	syl 14	. . . 4 ⊢ (𝜑 → 𝐾 ∈ SRing)
6	1, 2, 5	unitgrpbasd 14193	. . 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 14226	. . . 4 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐿))
12		eqidd 2232	. . . . 5 ⊢ (𝜑 → (Unit‘𝐿) = (Unit‘𝐿))
13		eqidd 2232	. . . . 5 ⊢ (𝜑 → ((mulGrp‘𝐿) ↾s (Unit‘𝐿)) = ((mulGrp‘𝐿) ↾s (Unit‘𝐿)))
14		ringsrg 14124	. . . . . 6 ⊢ (𝐿 ∈ Ring → 𝐿 ∈ SRing)
15	10, 14	syl 14	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ SRing)
16	12, 13, 15	unitgrpbasd 14193	. . . 4 ⊢ (𝜑 → (Unit‘𝐿) = (Base‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))))
17	11, 16	eqtrd 2264	. . 3 ⊢ (𝜑 → (Unit‘𝐾) = (Base‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))))
18		eqid 2231	. . . . . 6 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
19	18	ringmgp 14079	. . . . 5 ⊢ (𝐾 ∈ Ring → (mulGrp‘𝐾) ∈ Mnd)
20	3, 19	syl 14	. . . 4 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ Mnd)
21		basfn 13204	. . . . . . 7 ⊢ Base Fn V
22	3	elexd 2817	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ V)
23		funfvex 5665	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝐾 ∈ dom Base) → (Base‘𝐾) ∈ V)
24	23	funfni 5439	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝐾 ∈ V) → (Base‘𝐾) ∈ V)
25	21, 22, 24	sylancr 414	. . . . . 6 ⊢ (𝜑 → (Base‘𝐾) ∈ V)
26	7, 25	eqeltrd 2308	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
27	7, 1, 5	unitssd 14187	. . . . 5 ⊢ (𝜑 → (Unit‘𝐾) ⊆ 𝐵)
28	26, 27	ssexd 4234	. . . 4 ⊢ (𝜑 → (Unit‘𝐾) ∈ V)
29		ressex 13211	. . . 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 2231	. . . . . 6 ⊢ (mulGrp‘𝐿) = (mulGrp‘𝐿)
32	31	ringmgp 14079	. . . . 5 ⊢ (𝐿 ∈ Ring → (mulGrp‘𝐿) ∈ Mnd)
33	10, 32	syl 14	. . . 4 ⊢ (𝜑 → (mulGrp‘𝐿) ∈ Mnd)
34	11, 28	eqeltrrd 2309	. . . 4 ⊢ (𝜑 → (Unit‘𝐿) ∈ V)
35		ressex 13211	. . . 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 3228	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Unit‘𝐾)) → 𝑥 ∈ 𝐵)
38	27	sselda 3228	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (Unit‘𝐾)) → 𝑦 ∈ 𝐵)
39	37, 38	anim12dan 604	. . . . 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 2231	. . . . . . . 8 ⊢ (.r‘𝐾) = (.r‘𝐾)
42	18, 41	mgpplusgg 14001	. . . . . . 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 13275	. . . . 5 ⊢ (𝜑 → (.r‘𝐾) = (+g‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))))
45	44	oveqdr 6056	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Unit‘𝐾) ∧ 𝑦 ∈ (Unit‘𝐾))) → (𝑥(.r‘𝐾)𝑦) = (𝑥(+g‘((mulGrp‘𝐾) ↾s (Unit‘𝐾)))𝑦))
46		eqid 2231	. . . . . . . 8 ⊢ (.r‘𝐿) = (.r‘𝐿)
47	31, 46	mgpplusgg 14001	. . . . . . 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 13275	. . . . 5 ⊢ (𝜑 → (.r‘𝐿) = (+g‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))))
50	49	oveqdr 6056	. . . 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 2272	. . 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 13721	. 2 ⊢ (𝜑 → (invg‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))) = (invg‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))))
53		eqidd 2232	. . 3 ⊢ (𝜑 → (invr‘𝐾) = (invr‘𝐾))
54	1, 2, 53, 3	invrfvald 14200	. 2 ⊢ (𝜑 → (invr‘𝐾) = (invg‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))))
55		eqidd 2232	. . 3 ⊢ (𝜑 → (invr‘𝐿) = (invr‘𝐿))
56	12, 13, 55, 10	invrfvald 14200	. 2 ⊢ (𝜑 → (invr‘𝐿) = (invg‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))))
57	52, 54, 56	3eqtr4d 2274	1 ⊢ (𝜑 → (invr‘𝐾) = (invr‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2202  Vcvv 2803   Fn wfn 5328  ‘cfv 5333  (class class class)co 6028  Basecbs 13145   ↾_s cress 13146  +_gcplusg 13223  ._rcmulr 13224  Mndcmnd 13562  inv_gcminusg 13647  mulGrpcmgp 13997  SRingcsrg 14040  Ringcrg 14073  Unitcui 14164  inv_rcinvr 14198

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-pre-ltirr 8187  ax-pre-lttrn 8189  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-tpos 6454  df-pnf 8258  df-mnf 8259  df-ltxr 8261  df-inn 9186  df-2 9244  df-3 9245  df-ndx 13148  df-slot 13149  df-base 13151  df-sets 13152  df-iress 13153  df-plusg 13236  df-mulr 13237  df-0g 13404  df-mgm 13502  df-sgrp 13548  df-mnd 13563  df-grp 13649  df-minusg 13650  df-cmn 13936  df-abl 13937  df-mgp 13998  df-ur 14037  df-srg 14041  df-ring 14075  df-oppr 14145  df-dvdsr 14166  df-unit 14167  df-invr 14199

This theorem is referenced by: (None)