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Mirrors > Home > MPE Home > Th. List > ablsub2inv | Structured version Visualization version GIF version |
Description: Abelian group subtraction of two inverses. (Contributed by Stefan O'Rear, 24-May-2015.) |
Ref | Expression |
---|---|
ablsub2inv.b | ⊢ 𝐵 = (Base‘𝐺) |
ablsub2inv.m | ⊢ − = (-g‘𝐺) |
ablsub2inv.n | ⊢ 𝑁 = (invg‘𝐺) |
ablsub2inv.g | ⊢ (𝜑 → 𝐺 ∈ Abel) |
ablsub2inv.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
ablsub2inv.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
ablsub2inv | ⊢ (𝜑 → ((𝑁‘𝑋) − (𝑁‘𝑌)) = (𝑌 − 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ablsub2inv.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | eqid 2778 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | ablsub2inv.m | . . 3 ⊢ − = (-g‘𝐺) | |
4 | ablsub2inv.n | . . 3 ⊢ 𝑁 = (invg‘𝐺) | |
5 | ablsub2inv.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Abel) | |
6 | ablgrp 18588 | . . . 4 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Grp) |
8 | ablsub2inv.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
9 | 1, 4 | grpinvcl 17858 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵) |
10 | 7, 8, 9 | syl2anc 579 | . . 3 ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝐵) |
11 | ablsub2inv.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
12 | 1, 2, 3, 4, 7, 10, 11 | grpsubinv 17879 | . 2 ⊢ (𝜑 → ((𝑁‘𝑋) − (𝑁‘𝑌)) = ((𝑁‘𝑋)(+g‘𝐺)𝑌)) |
13 | 1, 2 | ablcom 18600 | . . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ (𝑁‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑁‘𝑋)(+g‘𝐺)𝑌) = (𝑌(+g‘𝐺)(𝑁‘𝑋))) |
14 | 5, 10, 11, 13 | syl3anc 1439 | . . . . 5 ⊢ (𝜑 → ((𝑁‘𝑋)(+g‘𝐺)𝑌) = (𝑌(+g‘𝐺)(𝑁‘𝑋))) |
15 | 1, 4 | grpinvinv 17873 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑁‘𝑌)) = 𝑌) |
16 | 7, 11, 15 | syl2anc 579 | . . . . . 6 ⊢ (𝜑 → (𝑁‘(𝑁‘𝑌)) = 𝑌) |
17 | 16 | oveq1d 6939 | . . . . 5 ⊢ (𝜑 → ((𝑁‘(𝑁‘𝑌))(+g‘𝐺)(𝑁‘𝑋)) = (𝑌(+g‘𝐺)(𝑁‘𝑋))) |
18 | 14, 17 | eqtr4d 2817 | . . . 4 ⊢ (𝜑 → ((𝑁‘𝑋)(+g‘𝐺)𝑌) = ((𝑁‘(𝑁‘𝑌))(+g‘𝐺)(𝑁‘𝑋))) |
19 | 1, 4 | grpinvcl 17858 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑁‘𝑌) ∈ 𝐵) |
20 | 7, 11, 19 | syl2anc 579 | . . . . 5 ⊢ (𝜑 → (𝑁‘𝑌) ∈ 𝐵) |
21 | 1, 2, 4 | grpinvadd 17884 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ (𝑁‘𝑌) ∈ 𝐵) → (𝑁‘(𝑋(+g‘𝐺)(𝑁‘𝑌))) = ((𝑁‘(𝑁‘𝑌))(+g‘𝐺)(𝑁‘𝑋))) |
22 | 7, 8, 20, 21 | syl3anc 1439 | . . . 4 ⊢ (𝜑 → (𝑁‘(𝑋(+g‘𝐺)(𝑁‘𝑌))) = ((𝑁‘(𝑁‘𝑌))(+g‘𝐺)(𝑁‘𝑋))) |
23 | 18, 22 | eqtr4d 2817 | . . 3 ⊢ (𝜑 → ((𝑁‘𝑋)(+g‘𝐺)𝑌) = (𝑁‘(𝑋(+g‘𝐺)(𝑁‘𝑌)))) |
24 | 1, 2, 4, 3 | grpsubval 17856 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋(+g‘𝐺)(𝑁‘𝑌))) |
25 | 8, 11, 24 | syl2anc 579 | . . . 4 ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑋(+g‘𝐺)(𝑁‘𝑌))) |
26 | 25 | fveq2d 6452 | . . 3 ⊢ (𝜑 → (𝑁‘(𝑋 − 𝑌)) = (𝑁‘(𝑋(+g‘𝐺)(𝑁‘𝑌)))) |
27 | 23, 26 | eqtr4d 2817 | . 2 ⊢ (𝜑 → ((𝑁‘𝑋)(+g‘𝐺)𝑌) = (𝑁‘(𝑋 − 𝑌))) |
28 | 1, 3, 4 | grpinvsub 17888 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑋 − 𝑌)) = (𝑌 − 𝑋)) |
29 | 7, 8, 11, 28 | syl3anc 1439 | . 2 ⊢ (𝜑 → (𝑁‘(𝑋 − 𝑌)) = (𝑌 − 𝑋)) |
30 | 12, 27, 29 | 3eqtrd 2818 | 1 ⊢ (𝜑 → ((𝑁‘𝑋) − (𝑁‘𝑌)) = (𝑌 − 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ‘cfv 6137 (class class class)co 6924 Basecbs 16259 +gcplusg 16342 Grpcgrp 17813 invgcminusg 17814 -gcsg 17815 Abelcabl 18584 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-1st 7447 df-2nd 7448 df-0g 16492 df-mgm 17632 df-sgrp 17674 df-mnd 17685 df-grp 17816 df-minusg 17817 df-sbg 17818 df-cmn 18585 df-abl 18586 |
This theorem is referenced by: ngpinvds 22829 |
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