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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 19803 | . . . 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 19005 | . . . 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 19030 | . 2 ⊢ (𝜑 → ((𝑁‘𝑋) − (𝑁‘𝑌)) = ((𝑁‘𝑋)(+g‘𝐺)𝑌)) | 
| 13 | 1, 2 | ablcom 19817 | . . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ (𝑁‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑁‘𝑋)(+g‘𝐺)𝑌) = (𝑌(+g‘𝐺)(𝑁‘𝑋))) | 
| 14 | 5, 10, 11, 13 | syl3anc 1373 | . . . . 5 ⊢ (𝜑 → ((𝑁‘𝑋)(+g‘𝐺)𝑌) = (𝑌(+g‘𝐺)(𝑁‘𝑋))) | 
| 15 | 1, 4 | grpinvinv 19023 | . . . . . . 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 2780 | . . . 4 ⊢ (𝜑 → ((𝑁‘𝑋)(+g‘𝐺)𝑌) = ((𝑁‘(𝑁‘𝑌))(+g‘𝐺)(𝑁‘𝑋))) | 
| 19 | 1, 4 | grpinvcl 19005 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑁‘𝑌) ∈ 𝐵) | 
| 20 | 7, 11, 19 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝑁‘𝑌) ∈ 𝐵) | 
| 21 | 1, 2, 4 | grpinvadd 19036 | . . . . 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 1373 | . . . 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 19003 | . . . . 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 6910 | . . 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 19040 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑋 − 𝑌)) = (𝑌 − 𝑋)) | 
| 29 | 7, 8, 11, 28 | syl3anc 1373 | . 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 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Grpcgrp 18951 invgcminusg 18952 -gcsg 18953 Abelcabl 19799 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-cmn 19800 df-abl 19801 | 
| This theorem is referenced by: ngpinvds 24626 | 
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