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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 2735 | . . 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 19818 | . . . 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 19018 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵) |
10 | 7, 8, 9 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝐵) |
11 | ablsub2inv.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
12 | 1, 2, 3, 4, 7, 10, 11 | grpsubinv 19043 | . 2 ⊢ (𝜑 → ((𝑁‘𝑋) − (𝑁‘𝑌)) = ((𝑁‘𝑋)(+g‘𝐺)𝑌)) |
13 | 1, 2 | ablcom 19832 | . . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ (𝑁‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑁‘𝑋)(+g‘𝐺)𝑌) = (𝑌(+g‘𝐺)(𝑁‘𝑋))) |
14 | 5, 10, 11, 13 | syl3anc 1370 | . . . . 5 ⊢ (𝜑 → ((𝑁‘𝑋)(+g‘𝐺)𝑌) = (𝑌(+g‘𝐺)(𝑁‘𝑋))) |
15 | 1, 4 | grpinvinv 19036 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑁‘𝑌)) = 𝑌) |
16 | 7, 11, 15 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (𝑁‘(𝑁‘𝑌)) = 𝑌) |
17 | 16 | oveq1d 7446 | . . . . 5 ⊢ (𝜑 → ((𝑁‘(𝑁‘𝑌))(+g‘𝐺)(𝑁‘𝑋)) = (𝑌(+g‘𝐺)(𝑁‘𝑋))) |
18 | 14, 17 | eqtr4d 2778 | . . . 4 ⊢ (𝜑 → ((𝑁‘𝑋)(+g‘𝐺)𝑌) = ((𝑁‘(𝑁‘𝑌))(+g‘𝐺)(𝑁‘𝑋))) |
19 | 1, 4 | grpinvcl 19018 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑁‘𝑌) ∈ 𝐵) |
20 | 7, 11, 19 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝑁‘𝑌) ∈ 𝐵) |
21 | 1, 2, 4 | grpinvadd 19049 | . . . . 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 1370 | . . . 4 ⊢ (𝜑 → (𝑁‘(𝑋(+g‘𝐺)(𝑁‘𝑌))) = ((𝑁‘(𝑁‘𝑌))(+g‘𝐺)(𝑁‘𝑋))) |
23 | 18, 22 | eqtr4d 2778 | . . 3 ⊢ (𝜑 → ((𝑁‘𝑋)(+g‘𝐺)𝑌) = (𝑁‘(𝑋(+g‘𝐺)(𝑁‘𝑌)))) |
24 | 1, 2, 4, 3 | grpsubval 19016 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋(+g‘𝐺)(𝑁‘𝑌))) |
25 | 8, 11, 24 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑋(+g‘𝐺)(𝑁‘𝑌))) |
26 | 25 | fveq2d 6911 | . . 3 ⊢ (𝜑 → (𝑁‘(𝑋 − 𝑌)) = (𝑁‘(𝑋(+g‘𝐺)(𝑁‘𝑌)))) |
27 | 23, 26 | eqtr4d 2778 | . 2 ⊢ (𝜑 → ((𝑁‘𝑋)(+g‘𝐺)𝑌) = (𝑁‘(𝑋 − 𝑌))) |
28 | 1, 3, 4 | grpinvsub 19053 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑋 − 𝑌)) = (𝑌 − 𝑋)) |
29 | 7, 8, 11, 28 | syl3anc 1370 | . 2 ⊢ (𝜑 → (𝑁‘(𝑋 − 𝑌)) = (𝑌 − 𝑋)) |
30 | 12, 27, 29 | 3eqtrd 2779 | 1 ⊢ (𝜑 → ((𝑁‘𝑋) − (𝑁‘𝑌)) = (𝑌 − 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 Grpcgrp 18964 invgcminusg 18965 -gcsg 18966 Abelcabl 19814 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-sbg 18969 df-cmn 19815 df-abl 19816 |
This theorem is referenced by: ngpinvds 24642 |
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