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Mirrors > Home > ILE Home > Th. List > grpsubadd0sub | 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 2193 | . . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
4 | grpsubid.m | . . . 4 ⊢ − = (-g‘𝐺) | |
5 | 1, 2, 3, 4 | grpsubval 13121 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋 + ((invg‘𝐺)‘𝑌))) |
6 | 5 | 3adant1 1017 | . 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 13158 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) = ( 0 − 𝑌)) |
9 | 8 | 3adant2 1018 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) = ( 0 − 𝑌)) |
10 | 9 | oveq2d 5935 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + ((invg‘𝐺)‘𝑌)) = (𝑋 + ( 0 − 𝑌))) |
11 | 6, 10 | eqtrd 2226 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋 + ( 0 − 𝑌))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 980 = wceq 1364 ∈ wcel 2164 ‘cfv 5255 (class class class)co 5919 Basecbs 12621 +gcplusg 12698 0gc0g 12870 Grpcgrp 13075 invgcminusg 13076 -gcsg 13077 |
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-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-cnex 7965 ax-resscn 7966 ax-1re 7968 ax-addrcl 7971 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-id 4325 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-inn 8985 df-2 9043 df-ndx 12624 df-slot 12625 df-base 12627 df-plusg 12711 df-0g 12872 df-mgm 12942 df-sgrp 12988 df-mnd 13001 df-grp 13078 df-minusg 13079 df-sbg 13080 |
This theorem is referenced by: (None) |
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