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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 19018 | . . . . 5 ⊢ (𝜑 → (𝑁‘𝑌) ∈ 𝐵) |
7 | grpcominv.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
8 | 1, 2, 3, 6, 5, 7 | grpassd 18975 | . . . 4 ⊢ (𝜑 → (((𝑁‘𝑌) + 𝑌) + 𝑋) = ((𝑁‘𝑌) + (𝑌 + 𝑋))) |
9 | eqid 2734 | . . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
10 | 1, 2, 9, 4, 3, 5 | grplinvd 19024 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘𝑌) + 𝑌) = (0g‘𝐺)) |
11 | 10 | oveq1d 7445 | . . . . 5 ⊢ (𝜑 → (((𝑁‘𝑌) + 𝑌) + 𝑋) = ((0g‘𝐺) + 𝑋)) |
12 | 1, 2, 9, 3, 7 | grplidd 18999 | . . . . 5 ⊢ (𝜑 → ((0g‘𝐺) + 𝑋) = 𝑋) |
13 | 11, 12 | eqtr2d 2775 | . . . 4 ⊢ (𝜑 → 𝑋 = (((𝑁‘𝑌) + 𝑌) + 𝑋)) |
14 | grpcominv.1 | . . . . 5 ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋)) | |
15 | 14 | oveq2d 7446 | . . . 4 ⊢ (𝜑 → ((𝑁‘𝑌) + (𝑋 + 𝑌)) = ((𝑁‘𝑌) + (𝑌 + 𝑋))) |
16 | 8, 13, 15 | 3eqtr4rd 2785 | . . 3 ⊢ (𝜑 → ((𝑁‘𝑌) + (𝑋 + 𝑌)) = 𝑋) |
17 | 1, 2, 3, 6, 7, 5 | grpassd 18975 | . . 3 ⊢ (𝜑 → (((𝑁‘𝑌) + 𝑋) + 𝑌) = ((𝑁‘𝑌) + (𝑋 + 𝑌))) |
18 | 1, 2, 4, 3, 7, 5 | grpasscan2d 42493 | . . 3 ⊢ (𝜑 → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = 𝑋) |
19 | 16, 17, 18 | 3eqtr4rd 2785 | . 2 ⊢ (𝜑 → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = (((𝑁‘𝑌) + 𝑋) + 𝑌)) |
20 | 1, 2, 3, 7, 6 | grpcld 18977 | . . 3 ⊢ (𝜑 → (𝑋 + (𝑁‘𝑌)) ∈ 𝐵) |
21 | 1, 2, 3, 6, 7 | grpcld 18977 | . . 3 ⊢ (𝜑 → ((𝑁‘𝑌) + 𝑋) ∈ 𝐵) |
22 | 1, 2 | grprcan 19003 | . . 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 1371 | . 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 1536 ∈ wcel 2105 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 +gcplusg 17297 0gc0g 17485 Grpcgrp 18963 invgcminusg 18964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-fv 6570 df-riota 7387 df-ov 7433 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-minusg 18967 |
This theorem is referenced by: grpcominv2 42495 |
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