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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 2736 | . . . . . . 7 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
3 | 1, 2 | grpinvcl 18795 | . . . . . 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 2736 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
6 | 1, 5 | grpcl 18753 | . . . . 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 3196 | . 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 18791 | . . 3 ⊢ − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦))) |
12 | 11 | fmpo 7997 | . 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 3063 × cxp 5630 ⟶wf 6490 ‘cfv 6494 (class class class)co 7354 Basecbs 17080 +gcplusg 17130 Grpcgrp 18745 invgcminusg 18746 -gcsg 18747 |
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 2707 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-fv 6502 df-riota 7310 df-ov 7357 df-oprab 7358 df-mpo 7359 df-1st 7918 df-2nd 7919 df-0g 17320 df-mgm 18494 df-sgrp 18543 df-mnd 18554 df-grp 18748 df-minusg 18749 df-sbg 18750 |
This theorem is referenced by: grpsubcl 18823 cnfldsub 20821 distgp 23446 indistgp 23447 clssubg 23456 tgphaus 23464 qustgplem 23468 nrmmetd 23926 isngp2 23949 isngp3 23950 ngpds 23956 ngptgp 23988 tngnm 24011 tngngp2 24012 rrxds 24753 |
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