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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 8888 | . . . 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 8889 | . . 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 18983 | . . 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 19004 | . . 3 ⊢ (𝐺 ∈ Grp → 𝑁:𝐵⟶𝐵) |
| 12 | 11 | adantr 480 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → 𝑁:𝐵⟶𝐵) |
| 13 | fcompt 7153 | . . 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 19007 | . . 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 7729 | 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 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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