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| Mirrors > Home > MPE Home > Th. List > Mathboxes > grpcominv1 | Structured version Visualization version GIF version | ||
| Description: If two elements commute, then they commute with each other's inverses (case of the first element commuting with the inverse of the second element). (Contributed by SN, 29-Jan-2025.) |
| Ref | Expression |
|---|---|
| grpcominv.b | ⊢ 𝐵 = (Base‘𝐺) |
| grpcominv.p | ⊢ + = (+g‘𝐺) |
| grpcominv.n | ⊢ 𝑁 = (invg‘𝐺) |
| grpcominv.g | ⊢ (𝜑 → 𝐺 ∈ Grp) |
| grpcominv.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| grpcominv.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
| grpcominv.1 | ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
| Ref | Expression |
|---|---|
| grpcominv1 | ⊢ (𝜑 → (𝑋 + (𝑁‘𝑌)) = ((𝑁‘𝑌) + 𝑋)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | grpcominv.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
| 2 | grpcominv.p | . . . . 5 ⊢ + = (+g‘𝐺) | |
| 3 | grpcominv.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ Grp) | |
| 4 | grpcominv.n | . . . . . 6 ⊢ 𝑁 = (invg‘𝐺) | |
| 5 | grpcominv.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
| 6 | 1, 4, 3, 5 | grpinvcld 19006 | . . . . 5 ⊢ (𝜑 → (𝑁‘𝑌) ∈ 𝐵) |
| 7 | grpcominv.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 8 | 1, 2, 3, 6, 5, 7 | grpassd 18963 | . . . 4 ⊢ (𝜑 → (((𝑁‘𝑌) + 𝑌) + 𝑋) = ((𝑁‘𝑌) + (𝑌 + 𝑋))) |
| 9 | eqid 2737 | . . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
| 10 | 1, 2, 9, 4, 3, 5 | grplinvd 19012 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘𝑌) + 𝑌) = (0g‘𝐺)) |
| 11 | 10 | oveq1d 7446 | . . . . 5 ⊢ (𝜑 → (((𝑁‘𝑌) + 𝑌) + 𝑋) = ((0g‘𝐺) + 𝑋)) |
| 12 | 1, 2, 9, 3, 7 | grplidd 18987 | . . . . 5 ⊢ (𝜑 → ((0g‘𝐺) + 𝑋) = 𝑋) |
| 13 | 11, 12 | eqtr2d 2778 | . . . 4 ⊢ (𝜑 → 𝑋 = (((𝑁‘𝑌) + 𝑌) + 𝑋)) |
| 14 | grpcominv.1 | . . . . 5 ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋)) | |
| 15 | 14 | oveq2d 7447 | . . . 4 ⊢ (𝜑 → ((𝑁‘𝑌) + (𝑋 + 𝑌)) = ((𝑁‘𝑌) + (𝑌 + 𝑋))) |
| 16 | 8, 13, 15 | 3eqtr4rd 2788 | . . 3 ⊢ (𝜑 → ((𝑁‘𝑌) + (𝑋 + 𝑌)) = 𝑋) |
| 17 | 1, 2, 3, 6, 7, 5 | grpassd 18963 | . . 3 ⊢ (𝜑 → (((𝑁‘𝑌) + 𝑋) + 𝑌) = ((𝑁‘𝑌) + (𝑋 + 𝑌))) |
| 18 | 1, 2, 4, 3, 7, 5 | grpasscan2d 42517 | . . 3 ⊢ (𝜑 → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = 𝑋) |
| 19 | 16, 17, 18 | 3eqtr4rd 2788 | . 2 ⊢ (𝜑 → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = (((𝑁‘𝑌) + 𝑋) + 𝑌)) |
| 20 | 1, 2, 3, 7, 6 | grpcld 18965 | . . 3 ⊢ (𝜑 → (𝑋 + (𝑁‘𝑌)) ∈ 𝐵) |
| 21 | 1, 2, 3, 6, 7 | grpcld 18965 | . . 3 ⊢ (𝜑 → ((𝑁‘𝑌) + 𝑋) ∈ 𝐵) |
| 22 | 1, 2 | grprcan 18991 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ ((𝑋 + (𝑁‘𝑌)) ∈ 𝐵 ∧ ((𝑁‘𝑌) + 𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (((𝑋 + (𝑁‘𝑌)) + 𝑌) = (((𝑁‘𝑌) + 𝑋) + 𝑌) ↔ (𝑋 + (𝑁‘𝑌)) = ((𝑁‘𝑌) + 𝑋))) |
| 23 | 3, 20, 21, 5, 22 | syl13anc 1374 | . 2 ⊢ (𝜑 → (((𝑋 + (𝑁‘𝑌)) + 𝑌) = (((𝑁‘𝑌) + 𝑋) + 𝑌) ↔ (𝑋 + (𝑁‘𝑌)) = ((𝑁‘𝑌) + 𝑋))) |
| 24 | 19, 23 | mpbid 232 | 1 ⊢ (𝜑 → (𝑋 + (𝑁‘𝑌)) = ((𝑁‘𝑌) + 𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 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-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-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-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-fv 6569 df-riota 7388 df-ov 7434 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 |
| This theorem is referenced by: grpcominv2 42519 |
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