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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 8429 | . . . 4 ⊢ (𝑋 ∈ (𝐵 ↑m 𝐼) → (𝐵 ∈ V ∧ 𝐼 ∈ V)) | |
2 | 1 | simprd 498 | . . 3 ⊢ (𝑋 ∈ (𝐵 ↑m 𝐼) → 𝐼 ∈ V) |
3 | 2 | adantl 484 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → 𝐼 ∈ V) |
4 | elmapi 8430 | . . 3 ⊢ (𝑋 ∈ (𝐵 ↑m 𝐼) → 𝑋:𝐼⟶𝐵) | |
5 | 4 | adantl 484 | . 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 18133 | . . 3 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵) |
9 | 8 | adantr 483 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → 0 ∈ 𝐵) |
10 | grpvlinv.n | . . . 4 ⊢ 𝑁 = (invg‘𝐺) | |
11 | 6, 10 | grpinvf 18152 | . . 3 ⊢ (𝐺 ∈ Grp → 𝑁:𝐵⟶𝐵) |
12 | 11 | adantr 483 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → 𝑁:𝐵⟶𝐵) |
13 | fcompt 6897 | . . 3 ⊢ ((𝑁:𝐵⟶𝐵 ∧ 𝑋:𝐼⟶𝐵) → (𝑁 ∘ 𝑋) = (𝑥 ∈ 𝐼 ↦ (𝑁‘(𝑋‘𝑥)))) | |
14 | 11, 4, 13 | syl2an 597 | . 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 18154 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵) → ((𝑁‘𝑦) + 𝑦) = 0 ) |
17 | 16 | adantlr 713 | . 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) ∧ 𝑦 ∈ 𝐵) → ((𝑁‘𝑦) + 𝑦) = 0 ) |
18 | 3, 5, 9, 12, 14, 17 | caofinvl 7438 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → ((𝑁 ∘ 𝑋) ∘f + 𝑋) = (𝐼 × { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 {csn 4569 ↦ cmpt 5148 × cxp 5555 ∘ ccom 5561 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ∘f cof 7409 ↑m cmap 8408 Basecbs 16485 +gcplusg 16567 0gc0g 16715 Grpcgrp 18105 invgcminusg 18106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-1st 7691 df-2nd 7692 df-map 8410 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 |
This theorem is referenced by: mendring 39799 |
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