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Mirrors > Home > MPE Home > Th. List > grpvlinv | Structured version Visualization version GIF version |
Description: Tuple-wise left inverse in groups. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
Ref | Expression |
---|---|
grpvlinv.b | ⊢ 𝐵 = (Base‘𝐺) |
grpvlinv.p | ⊢ + = (+g‘𝐺) |
grpvlinv.n | ⊢ 𝑁 = (invg‘𝐺) |
grpvlinv.z | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
grpvlinv | ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → ((𝑁 ∘ 𝑋) ∘f + 𝑋) = (𝐼 × { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmapex 8886 | . . . 4 ⊢ (𝑋 ∈ (𝐵 ↑m 𝐼) → (𝐵 ∈ V ∧ 𝐼 ∈ V)) | |
2 | 1 | simprd 495 | . . 3 ⊢ (𝑋 ∈ (𝐵 ↑m 𝐼) → 𝐼 ∈ V) |
3 | 2 | adantl 481 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → 𝐼 ∈ V) |
4 | elmapi 8887 | . . 3 ⊢ (𝑋 ∈ (𝐵 ↑m 𝐼) → 𝑋:𝐼⟶𝐵) | |
5 | 4 | adantl 481 | . 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 18995 | . . 3 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵) |
9 | 8 | adantr 480 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → 0 ∈ 𝐵) |
10 | grpvlinv.n | . . . 4 ⊢ 𝑁 = (invg‘𝐺) | |
11 | 6, 10 | grpinvf 19016 | . . 3 ⊢ (𝐺 ∈ Grp → 𝑁:𝐵⟶𝐵) |
12 | 11 | adantr 480 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → 𝑁:𝐵⟶𝐵) |
13 | fcompt 7152 | . . 3 ⊢ ((𝑁:𝐵⟶𝐵 ∧ 𝑋:𝐼⟶𝐵) → (𝑁 ∘ 𝑋) = (𝑥 ∈ 𝐼 ↦ (𝑁‘(𝑋‘𝑥)))) | |
14 | 11, 4, 13 | syl2an 596 | . 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 19019 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵) → ((𝑁‘𝑦) + 𝑦) = 0 ) |
17 | 16 | adantlr 715 | . 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) ∧ 𝑦 ∈ 𝐵) → ((𝑁‘𝑦) + 𝑦) = 0 ) |
18 | 3, 5, 9, 12, 14, 17 | caofinvl 7728 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → ((𝑁 ∘ 𝑋) ∘f + 𝑋) = (𝐼 × { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 Vcvv 3477 {csn 4630 ↦ cmpt 5230 × cxp 5686 ∘ ccom 5692 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ∘f cof 7694 ↑m cmap 8864 Basecbs 17244 +gcplusg 17297 0gc0g 17485 Grpcgrp 18963 invgcminusg 18964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-1st 8012 df-2nd 8013 df-map 8866 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-minusg 18967 |
This theorem is referenced by: mendring 43176 |
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