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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 2737 | . . 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 19574 | . . . 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 18805 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵) |
10 | 7, 8, 9 | syl2anc 585 | . . 3 ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝐵) |
11 | ablsub2inv.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
12 | 1, 2, 3, 4, 7, 10, 11 | grpsubinv 18827 | . 2 ⊢ (𝜑 → ((𝑁‘𝑋) − (𝑁‘𝑌)) = ((𝑁‘𝑋)(+g‘𝐺)𝑌)) |
13 | 1, 2 | ablcom 19588 | . . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ (𝑁‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑁‘𝑋)(+g‘𝐺)𝑌) = (𝑌(+g‘𝐺)(𝑁‘𝑋))) |
14 | 5, 10, 11, 13 | syl3anc 1372 | . . . . 5 ⊢ (𝜑 → ((𝑁‘𝑋)(+g‘𝐺)𝑌) = (𝑌(+g‘𝐺)(𝑁‘𝑋))) |
15 | 1, 4 | grpinvinv 18821 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑁‘𝑌)) = 𝑌) |
16 | 7, 11, 15 | syl2anc 585 | . . . . . 6 ⊢ (𝜑 → (𝑁‘(𝑁‘𝑌)) = 𝑌) |
17 | 16 | oveq1d 7377 | . . . . 5 ⊢ (𝜑 → ((𝑁‘(𝑁‘𝑌))(+g‘𝐺)(𝑁‘𝑋)) = (𝑌(+g‘𝐺)(𝑁‘𝑋))) |
18 | 14, 17 | eqtr4d 2780 | . . . 4 ⊢ (𝜑 → ((𝑁‘𝑋)(+g‘𝐺)𝑌) = ((𝑁‘(𝑁‘𝑌))(+g‘𝐺)(𝑁‘𝑋))) |
19 | 1, 4 | grpinvcl 18805 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑁‘𝑌) ∈ 𝐵) |
20 | 7, 11, 19 | syl2anc 585 | . . . . 5 ⊢ (𝜑 → (𝑁‘𝑌) ∈ 𝐵) |
21 | 1, 2, 4 | grpinvadd 18832 | . . . . 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 1372 | . . . 4 ⊢ (𝜑 → (𝑁‘(𝑋(+g‘𝐺)(𝑁‘𝑌))) = ((𝑁‘(𝑁‘𝑌))(+g‘𝐺)(𝑁‘𝑋))) |
23 | 18, 22 | eqtr4d 2780 | . . 3 ⊢ (𝜑 → ((𝑁‘𝑋)(+g‘𝐺)𝑌) = (𝑁‘(𝑋(+g‘𝐺)(𝑁‘𝑌)))) |
24 | 1, 2, 4, 3 | grpsubval 18803 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋(+g‘𝐺)(𝑁‘𝑌))) |
25 | 8, 11, 24 | syl2anc 585 | . . . 4 ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑋(+g‘𝐺)(𝑁‘𝑌))) |
26 | 25 | fveq2d 6851 | . . 3 ⊢ (𝜑 → (𝑁‘(𝑋 − 𝑌)) = (𝑁‘(𝑋(+g‘𝐺)(𝑁‘𝑌)))) |
27 | 23, 26 | eqtr4d 2780 | . 2 ⊢ (𝜑 → ((𝑁‘𝑋)(+g‘𝐺)𝑌) = (𝑁‘(𝑋 − 𝑌))) |
28 | 1, 3, 4 | grpinvsub 18836 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑋 − 𝑌)) = (𝑌 − 𝑋)) |
29 | 7, 8, 11, 28 | syl3anc 1372 | . 2 ⊢ (𝜑 → (𝑁‘(𝑋 − 𝑌)) = (𝑌 − 𝑋)) |
30 | 12, 27, 29 | 3eqtrd 2781 | 1 ⊢ (𝜑 → ((𝑁‘𝑋) − (𝑁‘𝑌)) = (𝑌 − 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ‘cfv 6501 (class class class)co 7362 Basecbs 17090 +gcplusg 17140 Grpcgrp 18755 invgcminusg 18756 -gcsg 18757 Abelcabl 19570 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-1st 7926 df-2nd 7927 df-0g 17330 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-grp 18758 df-minusg 18759 df-sbg 18760 df-cmn 19571 df-abl 19572 |
This theorem is referenced by: ngpinvds 23985 |
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