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Mirrors > Home > ILE Home > Th. List > grpsubfvalg | GIF version |
Description: Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 19-Feb-2024.) |
Ref | Expression |
---|---|
grpsubval.b | ⊢ 𝐵 = (Base‘𝐺) |
grpsubval.p | ⊢ + = (+g‘𝐺) |
grpsubval.i | ⊢ 𝐼 = (invg‘𝐺) |
grpsubval.m | ⊢ − = (-g‘𝐺) |
Ref | Expression |
---|---|
grpsubfvalg | ⊢ (𝐺 ∈ 𝑉 → − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpsubval.m | . 2 ⊢ − = (-g‘𝐺) | |
2 | df-sbg 12769 | . . 3 ⊢ -g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦)))) | |
3 | fveq2 5511 | . . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺)) | |
4 | grpsubval.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
5 | 3, 4 | eqtr4di 2228 | . . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵) |
6 | fveq2 5511 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺)) | |
7 | grpsubval.p | . . . . . 6 ⊢ + = (+g‘𝐺) | |
8 | 6, 7 | eqtr4di 2228 | . . . . 5 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + ) |
9 | eqidd 2178 | . . . . 5 ⊢ (𝑔 = 𝐺 → 𝑥 = 𝑥) | |
10 | fveq2 5511 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (invg‘𝑔) = (invg‘𝐺)) | |
11 | grpsubval.i | . . . . . . 7 ⊢ 𝐼 = (invg‘𝐺) | |
12 | 10, 11 | eqtr4di 2228 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (invg‘𝑔) = 𝐼) |
13 | 12 | fveq1d 5513 | . . . . 5 ⊢ (𝑔 = 𝐺 → ((invg‘𝑔)‘𝑦) = (𝐼‘𝑦)) |
14 | 8, 9, 13 | oveq123d 5890 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦)) = (𝑥 + (𝐼‘𝑦))) |
15 | 5, 5, 14 | mpoeq123dv 5931 | . . 3 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦))) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦)))) |
16 | elex 2748 | . . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V) | |
17 | basfn 12499 | . . . . . 6 ⊢ Base Fn V | |
18 | funfvex 5528 | . . . . . . 7 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V) | |
19 | 18 | funfni 5312 | . . . . . 6 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V) |
20 | 17, 16, 19 | sylancr 414 | . . . . 5 ⊢ (𝐺 ∈ 𝑉 → (Base‘𝐺) ∈ V) |
21 | 4, 20 | eqeltrid 2264 | . . . 4 ⊢ (𝐺 ∈ 𝑉 → 𝐵 ∈ V) |
22 | mpoexga 6207 | . . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) ∈ V) | |
23 | 21, 21, 22 | syl2anc 411 | . . 3 ⊢ (𝐺 ∈ 𝑉 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) ∈ V) |
24 | 2, 15, 16, 23 | fvmptd3 5605 | . 2 ⊢ (𝐺 ∈ 𝑉 → (-g‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦)))) |
25 | 1, 24 | eqtrid 2222 | 1 ⊢ (𝐺 ∈ 𝑉 → − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 Vcvv 2737 Fn wfn 5207 ‘cfv 5212 (class class class)co 5869 ∈ cmpo 5871 Basecbs 12442 +gcplusg 12515 invgcminusg 12765 -gcsg 12766 |
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-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 4206 ax-un 4430 ax-cnex 7890 ax-resscn 7891 ax-1re 7893 ax-addrcl 7896 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 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-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 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 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-inn 8906 df-ndx 12445 df-slot 12446 df-base 12448 df-sbg 12769 |
This theorem is referenced by: grpsubval 12806 grpsubf 12835 grpsubpropdg 12860 grpsubpropd2 12861 |
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