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Theorem grpvlinv 22402
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 + = (+g𝐺)
grpvlinv.n 𝑁 = (invg𝐺)
grpvlinv.z 0 = (0g𝐺)
Assertion
Ref Expression
grpvlinv ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵m 𝐼)) → ((𝑁𝑋) ∘f + 𝑋) = (𝐼 × { 0 }))

Proof of Theorem grpvlinv
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elmapex 8888 . . . 4 (𝑋 ∈ (𝐵m 𝐼) → (𝐵 ∈ V ∧ 𝐼 ∈ V))
21simprd 495 . . 3 (𝑋 ∈ (𝐵m 𝐼) → 𝐼 ∈ V)
32adantl 481 . 2 ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵m 𝐼)) → 𝐼 ∈ V)
4 elmapi 8889 . . 3 (𝑋 ∈ (𝐵m 𝐼) → 𝑋:𝐼𝐵)
54adantl 481 . 2 ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵m 𝐼)) → 𝑋:𝐼𝐵)
6 grpvlinv.b . . . 4 𝐵 = (Base‘𝐺)
7 grpvlinv.z . . . 4 0 = (0g𝐺)
86, 7grpidcl 18983 . . 3 (𝐺 ∈ Grp → 0𝐵)
98adantr 480 . 2 ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵m 𝐼)) → 0𝐵)
10 grpvlinv.n . . . 4 𝑁 = (invg𝐺)
116, 10grpinvf 19004 . . 3 (𝐺 ∈ Grp → 𝑁:𝐵𝐵)
1211adantr 480 . 2 ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵m 𝐼)) → 𝑁:𝐵𝐵)
13 fcompt 7153 . . 3 ((𝑁:𝐵𝐵𝑋:𝐼𝐵) → (𝑁𝑋) = (𝑥𝐼 ↦ (𝑁‘(𝑋𝑥))))
1411, 4, 13syl2an 596 . 2 ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵m 𝐼)) → (𝑁𝑋) = (𝑥𝐼 ↦ (𝑁‘(𝑋𝑥))))
15 grpvlinv.p . . . 4 + = (+g𝐺)
166, 15, 7, 10grplinv 19007 . . 3 ((𝐺 ∈ Grp ∧ 𝑦𝐵) → ((𝑁𝑦) + 𝑦) = 0 )
1716adantlr 715 . 2 (((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵m 𝐼)) ∧ 𝑦𝐵) → ((𝑁𝑦) + 𝑦) = 0 )
183, 5, 9, 12, 14, 17caofinvl 7729 1 ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵m 𝐼)) → ((𝑁𝑋) ∘f + 𝑋) = (𝐼 × { 0 }))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  {csn 4626  cmpt 5225   × cxp 5683  ccom 5689  wf 6557  cfv 6561  (class class class)co 7431  f cof 7695  m cmap 8866  Basecbs 17247  +gcplusg 17297  0gc0g 17484  Grpcgrp 18951  invgcminusg 18952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-1st 8014  df-2nd 8015  df-map 8868  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955
This theorem is referenced by:  mendring  43200
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