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Mirrors > Home > MPE Home > Th. List > Mathboxes > grpcominv2 | Structured version Visualization version GIF version |
Description: If two elements commute, then they commute with each other's inverses (case of the second element commuting with the inverse of the first element). (Contributed by SN, 1-Feb-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 |
---|---|
grpcominv2 | ⊢ (𝜑 → (𝑌 + (𝑁‘𝑋)) = ((𝑁‘𝑋) + 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpcominv.b | . 2 ⊢ 𝐵 = (Base‘𝐺) | |
2 | grpcominv.p | . 2 ⊢ + = (+g‘𝐺) | |
3 | grpcominv.n | . 2 ⊢ 𝑁 = (invg‘𝐺) | |
4 | grpcominv.g | . 2 ⊢ (𝜑 → 𝐺 ∈ Grp) | |
5 | grpcominv.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
6 | grpcominv.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
7 | grpcominv.1 | . . 3 ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋)) | |
8 | 7 | eqcomd 2731 | . 2 ⊢ (𝜑 → (𝑌 + 𝑋) = (𝑋 + 𝑌)) |
9 | 1, 2, 3, 4, 5, 6, 8 | grpcominv1 41804 | 1 ⊢ (𝜑 → (𝑌 + (𝑁‘𝑋)) = ((𝑁‘𝑋) + 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ‘cfv 6543 (class class class)co 7416 Basecbs 17179 +gcplusg 17232 Grpcgrp 18894 invgcminusg 18895 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-riota 7372 df-ov 7419 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18897 df-minusg 18898 |
This theorem is referenced by: (None) |
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