Theorem grpvrinv 20404

Description: Tuple-wise right inverse in groups. (Contributed by Mario Carneiro, 22-Sep-2015.)

Hypotheses

Ref	Expression
grpvlinv.b	⊢ 𝐵 = (Base‘𝐺)
grpvlinv.p	⊢ + = (+g‘𝐺)
grpvlinv.n	⊢ 𝑁 = (invg‘𝐺)
grpvlinv.z	⊢ 0 = (0g‘𝐺)

Assertion

Ref	Expression
grpvrinv	⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) → (𝑋 ∘𝑓 + (𝑁 ∘ 𝑋)) = (𝐼 × { 0 }))

Proof of Theorem grpvrinv

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 807	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) ∧ 𝑥 ∈ 𝐼) → 𝐺 ∈ Grp)
2		elmapi 8045	. . . . . 6 ⊢ (𝑋 ∈ (𝐵 ↑𝑚 𝐼) → 𝑋:𝐼⟶𝐵)
3	2	adantl 473	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) → 𝑋:𝐼⟶𝐵)
4	3	ffvelrnda 6522	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) ∧ 𝑥 ∈ 𝐼) → (𝑋‘𝑥) ∈ 𝐵)
5		grpvlinv.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
6		grpvlinv.p	. . . . 5 ⊢ + = (+g‘𝐺)
7		grpvlinv.z	. . . . 5 ⊢ 0 = (0g‘𝐺)
8		grpvlinv.n	. . . . 5 ⊢ 𝑁 = (invg‘𝐺)
9	5, 6, 7, 8	grprinv 17670	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑋‘𝑥) ∈ 𝐵) → ((𝑋‘𝑥) + (𝑁‘(𝑋‘𝑥))) = 0 )
10	1, 4, 9	syl2anc 696	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) ∧ 𝑥 ∈ 𝐼) → ((𝑋‘𝑥) + (𝑁‘(𝑋‘𝑥))) = 0 )
11	10	mpteq2dva 4896	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) → (𝑥 ∈ 𝐼 ↦ ((𝑋‘𝑥) + (𝑁‘(𝑋‘𝑥)))) = (𝑥 ∈ 𝐼 ↦ 0 ))
12		elmapex 8044	. . . . 5 ⊢ (𝑋 ∈ (𝐵 ↑𝑚 𝐼) → (𝐵 ∈ V ∧ 𝐼 ∈ V))
13	12	simprd 482	. . . 4 ⊢ (𝑋 ∈ (𝐵 ↑𝑚 𝐼) → 𝐼 ∈ V)
14	13	adantl 473	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) → 𝐼 ∈ V)
15		fvexd 6364	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) ∧ 𝑥 ∈ 𝐼) → (𝑁‘(𝑋‘𝑥)) ∈ V)
16	3	feqmptd 6411	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) → 𝑋 = (𝑥 ∈ 𝐼 ↦ (𝑋‘𝑥)))
17	5, 8	grpinvf 17667	. . . 4 ⊢ (𝐺 ∈ Grp → 𝑁:𝐵⟶𝐵)
18		fcompt 6563	. . . 4 ⊢ ((𝑁:𝐵⟶𝐵 ∧ 𝑋:𝐼⟶𝐵) → (𝑁 ∘ 𝑋) = (𝑥 ∈ 𝐼 ↦ (𝑁‘(𝑋‘𝑥))))
19	17, 2, 18	syl2an 495	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) → (𝑁 ∘ 𝑋) = (𝑥 ∈ 𝐼 ↦ (𝑁‘(𝑋‘𝑥))))
20	14, 4, 15, 16, 19	offval2 7079	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) → (𝑋 ∘𝑓 + (𝑁 ∘ 𝑋)) = (𝑥 ∈ 𝐼 ↦ ((𝑋‘𝑥) + (𝑁‘(𝑋‘𝑥)))))
21		fconstmpt 5320	. . 3 ⊢ (𝐼 × { 0 }) = (𝑥 ∈ 𝐼 ↦ 0 )
22	21	a1i 11	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) → (𝐼 × { 0 }) = (𝑥 ∈ 𝐼 ↦ 0 ))
23	11, 20, 22	3eqtr4d 2804	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) → (𝑋 ∘𝑓 + (𝑁 ∘ 𝑋)) = (𝐼 × { 0 }))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  Vcvv 3340  {csn 4321   ↦ cmpt 4881   × cxp 5264   ∘ ccom 5270  ⟶wf 6045  ‘cfv 6049  (class class class)co 6813   ∘_𝑓 cof 7060   ↑_𝑚 cmap 8023  Basecbs 16059  +_gcplusg 16143  0_gc0g 16302  Grpcgrp 17623  inv_gcminusg 17624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-1st 7333  df-2nd 7334  df-map 8025  df-0g 16304  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-grp 17626  df-minusg 17627

This theorem is referenced by: (None)