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Mirrors > Home > MPE Home > Th. List > grpsubf | Structured version Visualization version 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 | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐺) | |
2 | eqid 2732 | . . . . . . 7 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
3 | 1, 2 | grpinvcl 18871 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵) → ((invg‘𝐺)‘𝑦) ∈ 𝐵) |
4 | 3 | 3adant2 1131 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((invg‘𝐺)‘𝑦) ∈ 𝐵) |
5 | eqid 2732 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
6 | 1, 5 | grpcl 18826 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ ((invg‘𝐺)‘𝑦) ∈ 𝐵) → (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦)) ∈ 𝐵) |
7 | 4, 6 | syld3an3 1409 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦)) ∈ 𝐵) |
8 | 7 | 3expb 1120 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦)) ∈ 𝐵) |
9 | 8 | ralrimivva 3200 | . 2 ⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦)) ∈ 𝐵) |
10 | grpsubcl.m | . . . 4 ⊢ − = (-g‘𝐺) | |
11 | 1, 5, 2, 10 | grpsubfval 18867 | . . 3 ⊢ − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦))) |
12 | 11 | fmpo 8053 | . 2 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦)) ∈ 𝐵 ↔ − :(𝐵 × 𝐵)⟶𝐵) |
13 | 9, 12 | sylib 217 | 1 ⊢ (𝐺 ∈ Grp → − :(𝐵 × 𝐵)⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ∀wral 3061 × cxp 5674 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 Basecbs 17143 +gcplusg 17196 Grpcgrp 18818 invgcminusg 18819 -gcsg 18820 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-0g 17386 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-grp 18821 df-minusg 18822 df-sbg 18823 |
This theorem is referenced by: grpsubcl 18902 cnfldsub 20972 distgp 23602 indistgp 23603 clssubg 23612 tgphaus 23620 qustgplem 23624 nrmmetd 24082 isngp2 24105 isngp3 24106 ngpds 24112 ngptgp 24144 tngnm 24167 tngngp2 24168 rrxds 24909 |
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