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Mirrors > Home > MPE Home > Th. List > grpsubadd0sub | Structured version Visualization version GIF version |
Description: Subtraction expressed as addition of the difference of the identity element and the subtrahend. (Contributed by AV, 9-Nov-2019.) |
Ref | Expression |
---|---|
grpsubid.b | ⊢ 𝐵 = (Base‘𝐺) |
grpsubid.o | ⊢ 0 = (0g‘𝐺) |
grpsubid.m | ⊢ − = (-g‘𝐺) |
grpsubadd0sub.p | ⊢ + = (+g‘𝐺) |
Ref | Expression |
---|---|
grpsubadd0sub | ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋 + ( 0 − 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpsubid.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
2 | grpsubadd0sub.p | . . . 4 ⊢ + = (+g‘𝐺) | |
3 | eqid 2733 | . . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
4 | grpsubid.m | . . . 4 ⊢ − = (-g‘𝐺) | |
5 | 1, 2, 3, 4 | grpsubval 18801 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋 + ((invg‘𝐺)‘𝑌))) |
6 | 5 | 3adant1 1131 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋 + ((invg‘𝐺)‘𝑌))) |
7 | grpsubid.o | . . . . 5 ⊢ 0 = (0g‘𝐺) | |
8 | 1, 4, 3, 7 | grpinvval2 18835 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) = ( 0 − 𝑌)) |
9 | 8 | 3adant2 1132 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) = ( 0 − 𝑌)) |
10 | 9 | oveq2d 7374 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + ((invg‘𝐺)‘𝑌)) = (𝑋 + ( 0 − 𝑌))) |
11 | 6, 10 | eqtrd 2773 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋 + ( 0 − 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ‘cfv 6497 (class class class)co 7358 Basecbs 17088 +gcplusg 17138 0gc0g 17326 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: chfacfscmulgsum 22225 chfacfpmmulgsum 22229 |
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