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Mirrors > Home > ILE Home > Th. List > grpinvsub | GIF version |
Description: Inverse of a group subtraction. (Contributed by NM, 9-Sep-2014.) |
Ref | Expression |
---|---|
grpsubcl.b | ⊢ 𝐵 = (Base‘𝐺) |
grpsubcl.m | ⊢ − = (-g‘𝐺) |
grpinvsub.n | ⊢ 𝑁 = (invg‘𝐺) |
Ref | Expression |
---|---|
grpinvsub | ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑋 − 𝑌)) = (𝑌 − 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpsubcl.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
2 | grpinvsub.n | . . . . . 6 ⊢ 𝑁 = (invg‘𝐺) | |
3 | 1, 2 | grpinvcl 12798 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑁‘𝑌) ∈ 𝐵) |
4 | 3 | 3adant2 1016 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘𝑌) ∈ 𝐵) |
5 | eqid 2177 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
6 | 1, 5, 2 | grpinvadd 12824 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ (𝑁‘𝑌) ∈ 𝐵) → (𝑁‘(𝑋(+g‘𝐺)(𝑁‘𝑌))) = ((𝑁‘(𝑁‘𝑌))(+g‘𝐺)(𝑁‘𝑋))) |
7 | 4, 6 | syld3an3 1283 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑋(+g‘𝐺)(𝑁‘𝑌))) = ((𝑁‘(𝑁‘𝑌))(+g‘𝐺)(𝑁‘𝑋))) |
8 | 1, 2 | grpinvinv 12813 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑁‘𝑌)) = 𝑌) |
9 | 8 | 3adant2 1016 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑁‘𝑌)) = 𝑌) |
10 | 9 | oveq1d 5883 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑁‘(𝑁‘𝑌))(+g‘𝐺)(𝑁‘𝑋)) = (𝑌(+g‘𝐺)(𝑁‘𝑋))) |
11 | 7, 10 | eqtrd 2210 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑋(+g‘𝐺)(𝑁‘𝑌))) = (𝑌(+g‘𝐺)(𝑁‘𝑋))) |
12 | grpsubcl.m | . . . . 5 ⊢ − = (-g‘𝐺) | |
13 | 1, 5, 2, 12 | grpsubval 12796 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋(+g‘𝐺)(𝑁‘𝑌))) |
14 | 13 | 3adant1 1015 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋(+g‘𝐺)(𝑁‘𝑌))) |
15 | 14 | fveq2d 5514 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑋 − 𝑌)) = (𝑁‘(𝑋(+g‘𝐺)(𝑁‘𝑌)))) |
16 | 1, 5, 2, 12 | grpsubval 12796 | . . . 4 ⊢ ((𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑌 − 𝑋) = (𝑌(+g‘𝐺)(𝑁‘𝑋))) |
17 | 16 | ancoms 268 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 − 𝑋) = (𝑌(+g‘𝐺)(𝑁‘𝑋))) |
18 | 17 | 3adant1 1015 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 − 𝑋) = (𝑌(+g‘𝐺)(𝑁‘𝑋))) |
19 | 11, 15, 18 | 3eqtr4d 2220 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑋 − 𝑌)) = (𝑌 − 𝑋)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ‘cfv 5211 (class class class)co 5868 Basecbs 12432 +gcplusg 12505 Grpcgrp 12754 invgcminusg 12755 -gcsg 12756 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1re 7883 ax-addrcl 7886 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-1st 6134 df-2nd 6135 df-inn 8896 df-2 8954 df-ndx 12435 df-slot 12436 df-base 12438 df-plusg 12518 df-0g 12642 df-mgm 12654 df-sgrp 12687 df-mnd 12697 df-grp 12757 df-minusg 12758 df-sbg 12759 |
This theorem is referenced by: grpsubsub 12835 ablsub2inv 12928 |
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