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Mirrors > Home > MPE Home > Th. List > grpvrinv | Structured version Visualization version GIF version |
Description: Tuple-wise right inverse in groups. (Contributed by Mario Carneiro, 22-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 |
---|---|
grpvrinv | ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) → (𝑋 ∘𝑓 + (𝑁 ∘ 𝑋)) = (𝐼 × { 0 })) |
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 })) |
Colors of variables: wff setvar class |
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 0gc0g 16302 Grpcgrp 17623 invgcminusg 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) |
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