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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 786 | . . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) ∧ 𝑥 ∈ 𝐼) → 𝐺 ∈ Grp) | |
2 | elmapi 7765 | . . . . . 6 ⊢ (𝑋 ∈ (𝐵 ↑𝑚 𝐼) → 𝑋:𝐼⟶𝐵) | |
3 | 2 | adantl 481 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) → 𝑋:𝐼⟶𝐵) |
4 | 3 | ffvelrnda 6267 | . . . 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 17292 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑋‘𝑥) ∈ 𝐵) → ((𝑋‘𝑥) + (𝑁‘(𝑋‘𝑥))) = 0 ) |
10 | 1, 4, 9 | syl2anc 691 | . . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) ∧ 𝑥 ∈ 𝐼) → ((𝑋‘𝑥) + (𝑁‘(𝑋‘𝑥))) = 0 ) |
11 | 10 | mpteq2dva 4672 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) → (𝑥 ∈ 𝐼 ↦ ((𝑋‘𝑥) + (𝑁‘(𝑋‘𝑥)))) = (𝑥 ∈ 𝐼 ↦ 0 )) |
12 | elmapex 7764 | . . . . 5 ⊢ (𝑋 ∈ (𝐵 ↑𝑚 𝐼) → (𝐵 ∈ V ∧ 𝐼 ∈ V)) | |
13 | 12 | simprd 478 | . . . 4 ⊢ (𝑋 ∈ (𝐵 ↑𝑚 𝐼) → 𝐼 ∈ V) |
14 | 13 | adantl 481 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) → 𝐼 ∈ V) |
15 | fvex 6113 | . . . 4 ⊢ (𝑁‘(𝑋‘𝑥)) ∈ V | |
16 | 15 | a1i 11 | . . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) ∧ 𝑥 ∈ 𝐼) → (𝑁‘(𝑋‘𝑥)) ∈ V) |
17 | 3 | feqmptd 6159 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) → 𝑋 = (𝑥 ∈ 𝐼 ↦ (𝑋‘𝑥))) |
18 | 5, 8 | grpinvf 17289 | . . . 4 ⊢ (𝐺 ∈ Grp → 𝑁:𝐵⟶𝐵) |
19 | fcompt 6306 | . . . 4 ⊢ ((𝑁:𝐵⟶𝐵 ∧ 𝑋:𝐼⟶𝐵) → (𝑁 ∘ 𝑋) = (𝑥 ∈ 𝐼 ↦ (𝑁‘(𝑋‘𝑥)))) | |
20 | 18, 2, 19 | syl2an 493 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) → (𝑁 ∘ 𝑋) = (𝑥 ∈ 𝐼 ↦ (𝑁‘(𝑋‘𝑥)))) |
21 | 14, 4, 16, 17, 20 | offval2 6812 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) → (𝑋 ∘𝑓 + (𝑁 ∘ 𝑋)) = (𝑥 ∈ 𝐼 ↦ ((𝑋‘𝑥) + (𝑁‘(𝑋‘𝑥))))) |
22 | fconstmpt 5085 | . . 3 ⊢ (𝐼 × { 0 }) = (𝑥 ∈ 𝐼 ↦ 0 ) | |
23 | 22 | a1i 11 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) → (𝐼 × { 0 }) = (𝑥 ∈ 𝐼 ↦ 0 )) |
24 | 11, 21, 23 | 3eqtr4d 2654 | 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 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 ↦ cmpt 4643 × cxp 5036 ∘ ccom 5042 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ∘𝑓 cof 6793 ↑𝑚 cmap 7744 Basecbs 15695 +gcplusg 15768 0gc0g 15923 Grpcgrp 17245 invgcminusg 17246 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-1st 7059 df-2nd 7060 df-map 7746 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 |
This theorem is referenced by: (None) |
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