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Mirrors > Home > ILE Home > Th. List > grpsubf | GIF version |
Description: Functionality of group subtraction. (Contributed by Mario Carneiro, 9-Sep-2014.) |
Ref | Expression |
---|---|
grpsubcl.b | ⊢ 𝐵 = (Base‘𝐺) |
grpsubcl.m | ⊢ − = (-g‘𝐺) |
Ref | Expression |
---|---|
grpsubf | ⊢ (𝐺 ∈ Grp → − :(𝐵 × 𝐵)⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpsubcl.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝐺) | |
2 | eqid 2177 | . . . . . . . 8 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
3 | 1, 2 | grpinvcl 12811 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵) → ((invg‘𝐺)‘𝑦) ∈ 𝐵) |
4 | 3 | 3adant2 1016 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((invg‘𝐺)‘𝑦) ∈ 𝐵) |
5 | eqid 2177 | . . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
6 | 1, 5 | grpcl 12775 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ ((invg‘𝐺)‘𝑦) ∈ 𝐵) → (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦)) ∈ 𝐵) |
7 | 4, 6 | syld3an3 1283 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦)) ∈ 𝐵) |
8 | 7 | 3expb 1204 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦)) ∈ 𝐵) |
9 | 8 | ralrimivva 2559 | . . 3 ⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦)) ∈ 𝐵) |
10 | eqid 2177 | . . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦))) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦))) | |
11 | 10 | fmpo 6196 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦)) ∈ 𝐵 ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦))):(𝐵 × 𝐵)⟶𝐵) |
12 | 9, 11 | sylib 122 | . 2 ⊢ (𝐺 ∈ Grp → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦))):(𝐵 × 𝐵)⟶𝐵) |
13 | grpsubcl.m | . . . 4 ⊢ − = (-g‘𝐺) | |
14 | 1, 5, 2, 13 | grpsubfvalg 12808 | . . 3 ⊢ (𝐺 ∈ Grp → − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦)))) |
15 | 14 | feq1d 5348 | . 2 ⊢ (𝐺 ∈ Grp → ( − :(𝐵 × 𝐵)⟶𝐵 ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦))):(𝐵 × 𝐵)⟶𝐵)) |
16 | 12, 15 | mpbird 167 | 1 ⊢ (𝐺 ∈ Grp → − :(𝐵 × 𝐵)⟶𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 ∀wral 2455 × cxp 4621 ⟶wf 5208 ‘cfv 5212 (class class class)co 5869 ∈ cmpo 5871 Basecbs 12445 +gcplusg 12518 Grpcgrp 12767 invgcminusg 12768 -gcsg 12769 |
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 7893 ax-resscn 7894 ax-1re 7896 ax-addrcl 7899 |
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-rmo 2463 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-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-inn 8909 df-2 8967 df-ndx 12448 df-slot 12449 df-base 12451 df-plusg 12531 df-0g 12655 df-mgm 12667 df-sgrp 12700 df-mnd 12710 df-grp 12770 df-minusg 12771 df-sbg 12772 |
This theorem is referenced by: grpsubcl 12839 |
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