Theorem grpvlinv 20201

Description: Tuple-wise left inverse in groups. (Contributed by Stefan O'Rear, 5-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
grpvlinv	⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) → ((𝑁 ∘ 𝑋) ∘𝑓 + 𝑋) = (𝐼 × { 0 }))

Proof of Theorem grpvlinv

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elmapex 7878	. . . 4 ⊢ (𝑋 ∈ (𝐵 ↑𝑚 𝐼) → (𝐵 ∈ V ∧ 𝐼 ∈ V))
2	1	simprd 479	. . 3 ⊢ (𝑋 ∈ (𝐵 ↑𝑚 𝐼) → 𝐼 ∈ V)
3	2	adantl 482	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) → 𝐼 ∈ V)
4		elmapi 7879	. . 3 ⊢ (𝑋 ∈ (𝐵 ↑𝑚 𝐼) → 𝑋:𝐼⟶𝐵)
5	4	adantl 482	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) → 𝑋:𝐼⟶𝐵)
6		grpvlinv.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
7		grpvlinv.z	. . . 4 ⊢ 0 = (0g‘𝐺)
8	6, 7	grpidcl 17450	. . 3 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵)
9	8	adantr 481	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) → 0 ∈ 𝐵)
10		grpvlinv.n	. . . 4 ⊢ 𝑁 = (invg‘𝐺)
11	6, 10	grpinvf 17466	. . 3 ⊢ (𝐺 ∈ Grp → 𝑁:𝐵⟶𝐵)
12	11	adantr 481	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) → 𝑁:𝐵⟶𝐵)
13		fcompt 6400	. . 3 ⊢ ((𝑁:𝐵⟶𝐵 ∧ 𝑋:𝐼⟶𝐵) → (𝑁 ∘ 𝑋) = (𝑥 ∈ 𝐼 ↦ (𝑁‘(𝑋‘𝑥))))
14	11, 4, 13	syl2an 494	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) → (𝑁 ∘ 𝑋) = (𝑥 ∈ 𝐼 ↦ (𝑁‘(𝑋‘𝑥))))
15		grpvlinv.p	. . . 4 ⊢ + = (+g‘𝐺)
16	6, 15, 7, 10	grplinv 17468	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵) → ((𝑁‘𝑦) + 𝑦) = 0 )
17	16	adantlr 751	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) ∧ 𝑦 ∈ 𝐵) → ((𝑁‘𝑦) + 𝑦) = 0 )
18	3, 5, 9, 12, 14, 17	caofinvl 6924	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) → ((𝑁 ∘ 𝑋) ∘𝑓 + 𝑋) = (𝐼 × { 0 }))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  {csn 4177   ↦ cmpt 4729   × cxp 5112   ∘ ccom 5118  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ∘_𝑓 cof 6895   ↑_𝑚 cmap 7857  Basecbs 15857  +_gcplusg 15941  0_gc0g 16100  Grpcgrp 17422  inv_gcminusg 17423

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-1st 7168  df-2nd 7169  df-map 7859  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426

This theorem is referenced by:  mendring  37762