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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 2733 | . . . . . . 7 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
3 | 1, 2 | grpinvcl 18803 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵) → ((invg‘𝐺)‘𝑦) ∈ 𝐵) |
4 | 3 | 3adant2 1132 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((invg‘𝐺)‘𝑦) ∈ 𝐵) |
5 | eqid 2733 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
6 | 1, 5 | grpcl 18761 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ ((invg‘𝐺)‘𝑦) ∈ 𝐵) → (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦)) ∈ 𝐵) |
7 | 4, 6 | syld3an3 1410 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦)) ∈ 𝐵) |
8 | 7 | 3expb 1121 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦)) ∈ 𝐵) |
9 | 8 | ralrimivva 3194 | . 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 18799 | . . 3 ⊢ − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝐺)((invg‘𝐺)‘𝑦))) |
12 | 11 | fmpo 8001 | . 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 1542 ∈ wcel 2107 ∀wral 3061 × cxp 5632 ⟶wf 6493 ‘cfv 6497 (class class class)co 7358 Basecbs 17088 +gcplusg 17138 Grpcgrp 18753 invgcminusg 18754 -gcsg 18755 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-minusg 18757 df-sbg 18758 |
This theorem is referenced by: grpsubcl 18832 cnfldsub 20841 distgp 23466 indistgp 23467 clssubg 23476 tgphaus 23484 qustgplem 23488 nrmmetd 23946 isngp2 23969 isngp3 23970 ngpds 23976 ngptgp 24008 tngnm 24031 tngngp2 24032 rrxds 24773 |
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