Theorem grpvrinv 21009

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 765	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) ∧ 𝑥 ∈ 𝐼) → 𝐺 ∈ Grp)
2		elmapi 8430	. . . . . 6 ⊢ (𝑋 ∈ (𝐵 ↑m 𝐼) → 𝑋:𝐼⟶𝐵)
3	2	adantl 484	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → 𝑋:𝐼⟶𝐵)
4	3	ffvelrnda 6853	. . . 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 18155	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑋‘𝑥) ∈ 𝐵) → ((𝑋‘𝑥) + (𝑁‘(𝑋‘𝑥))) = 0 )
10	1, 4, 9	syl2anc 586	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) ∧ 𝑥 ∈ 𝐼) → ((𝑋‘𝑥) + (𝑁‘(𝑋‘𝑥))) = 0 )
11	10	mpteq2dva 5163	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → (𝑥 ∈ 𝐼 ↦ ((𝑋‘𝑥) + (𝑁‘(𝑋‘𝑥)))) = (𝑥 ∈ 𝐼 ↦ 0 ))
12		elmapex 8429	. . . . 5 ⊢ (𝑋 ∈ (𝐵 ↑m 𝐼) → (𝐵 ∈ V ∧ 𝐼 ∈ V))
13	12	simprd 498	. . . 4 ⊢ (𝑋 ∈ (𝐵 ↑m 𝐼) → 𝐼 ∈ V)
14	13	adantl 484	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → 𝐼 ∈ V)
15		fvexd 6687	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) ∧ 𝑥 ∈ 𝐼) → (𝑁‘(𝑋‘𝑥)) ∈ V)
16	3	feqmptd 6735	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → 𝑋 = (𝑥 ∈ 𝐼 ↦ (𝑋‘𝑥)))
17	5, 8	grpinvf 18152	. . . 4 ⊢ (𝐺 ∈ Grp → 𝑁:𝐵⟶𝐵)
18		fcompt 6897	. . . 4 ⊢ ((𝑁:𝐵⟶𝐵 ∧ 𝑋:𝐼⟶𝐵) → (𝑁 ∘ 𝑋) = (𝑥 ∈ 𝐼 ↦ (𝑁‘(𝑋‘𝑥))))
19	17, 2, 18	syl2an 597	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → (𝑁 ∘ 𝑋) = (𝑥 ∈ 𝐼 ↦ (𝑁‘(𝑋‘𝑥))))
20	14, 4, 15, 16, 19	offval2 7428	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → (𝑋 ∘f + (𝑁 ∘ 𝑋)) = (𝑥 ∈ 𝐼 ↦ ((𝑋‘𝑥) + (𝑁‘(𝑋‘𝑥)))))
21		fconstmpt 5616	. . 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 2868	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → (𝑋 ∘f + (𝑁 ∘ 𝑋)) = (𝐼 × { 0 }))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3496  {csn 4569   ↦ cmpt 5148   × cxp 5555   ∘ ccom 5561  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158   ∘_f cof 7409   ↑_m cmap 8408  Basecbs 16485  +_gcplusg 16567  0_gc0g 16715  Grpcgrp 18105  inv_gcminusg 18106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-of 7411  df-1st 7691  df-2nd 7692  df-map 8410  df-0g 16717  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-grp 18108  df-minusg 18109

This theorem is referenced by: (None)