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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 2771 | . . 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 18405 | . . . 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 17675 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵) |
10 | 7, 8, 9 | syl2anc 573 | . . 3 ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝐵) |
11 | ablsub2inv.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
12 | 1, 2, 3, 4, 7, 10, 11 | grpsubinv 17696 | . 2 ⊢ (𝜑 → ((𝑁‘𝑋) − (𝑁‘𝑌)) = ((𝑁‘𝑋)(+g‘𝐺)𝑌)) |
13 | 1, 2 | ablcom 18417 | . . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ (𝑁‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑁‘𝑋)(+g‘𝐺)𝑌) = (𝑌(+g‘𝐺)(𝑁‘𝑋))) |
14 | 5, 10, 11, 13 | syl3anc 1476 | . . . . 5 ⊢ (𝜑 → ((𝑁‘𝑋)(+g‘𝐺)𝑌) = (𝑌(+g‘𝐺)(𝑁‘𝑋))) |
15 | 1, 4 | grpinvinv 17690 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑁‘𝑌)) = 𝑌) |
16 | 7, 11, 15 | syl2anc 573 | . . . . . 6 ⊢ (𝜑 → (𝑁‘(𝑁‘𝑌)) = 𝑌) |
17 | 16 | oveq1d 6811 | . . . . 5 ⊢ (𝜑 → ((𝑁‘(𝑁‘𝑌))(+g‘𝐺)(𝑁‘𝑋)) = (𝑌(+g‘𝐺)(𝑁‘𝑋))) |
18 | 14, 17 | eqtr4d 2808 | . . . 4 ⊢ (𝜑 → ((𝑁‘𝑋)(+g‘𝐺)𝑌) = ((𝑁‘(𝑁‘𝑌))(+g‘𝐺)(𝑁‘𝑋))) |
19 | 1, 4 | grpinvcl 17675 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑁‘𝑌) ∈ 𝐵) |
20 | 7, 11, 19 | syl2anc 573 | . . . . 5 ⊢ (𝜑 → (𝑁‘𝑌) ∈ 𝐵) |
21 | 1, 2, 4 | grpinvadd 17701 | . . . . 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 1476 | . . . 4 ⊢ (𝜑 → (𝑁‘(𝑋(+g‘𝐺)(𝑁‘𝑌))) = ((𝑁‘(𝑁‘𝑌))(+g‘𝐺)(𝑁‘𝑋))) |
23 | 18, 22 | eqtr4d 2808 | . . 3 ⊢ (𝜑 → ((𝑁‘𝑋)(+g‘𝐺)𝑌) = (𝑁‘(𝑋(+g‘𝐺)(𝑁‘𝑌)))) |
24 | 1, 2, 4, 3 | grpsubval 17673 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋(+g‘𝐺)(𝑁‘𝑌))) |
25 | 8, 11, 24 | syl2anc 573 | . . . 4 ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑋(+g‘𝐺)(𝑁‘𝑌))) |
26 | 25 | fveq2d 6337 | . . 3 ⊢ (𝜑 → (𝑁‘(𝑋 − 𝑌)) = (𝑁‘(𝑋(+g‘𝐺)(𝑁‘𝑌)))) |
27 | 23, 26 | eqtr4d 2808 | . 2 ⊢ (𝜑 → ((𝑁‘𝑋)(+g‘𝐺)𝑌) = (𝑁‘(𝑋 − 𝑌))) |
28 | 1, 3, 4 | grpinvsub 17705 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑋 − 𝑌)) = (𝑌 − 𝑋)) |
29 | 7, 8, 11, 28 | syl3anc 1476 | . 2 ⊢ (𝜑 → (𝑁‘(𝑋 − 𝑌)) = (𝑌 − 𝑋)) |
30 | 12, 27, 29 | 3eqtrd 2809 | 1 ⊢ (𝜑 → ((𝑁‘𝑋) − (𝑁‘𝑌)) = (𝑌 − 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 ‘cfv 6030 (class class class)co 6796 Basecbs 16064 +gcplusg 16149 Grpcgrp 17630 invgcminusg 17631 -gcsg 17632 Abelcabl 18401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-1st 7319 df-2nd 7320 df-0g 16310 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-grp 17633 df-minusg 17634 df-sbg 17635 df-cmn 18402 df-abl 18403 |
This theorem is referenced by: ngpinvds 22637 |
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