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Mirrors > Home > MPE Home > Th. List > ablsubadd | Structured version Visualization version GIF version |
Description: Relationship between Abelian group subtraction and addition. (Contributed by NM, 31-Mar-2014.) |
Ref | Expression |
---|---|
ablsubadd.b | ⊢ 𝐵 = (Base‘𝐺) |
ablsubadd.p | ⊢ + = (+g‘𝐺) |
ablsubadd.m | ⊢ − = (-g‘𝐺) |
Ref | Expression |
---|---|
ablsubadd | ⊢ ((𝐺 ∈ Abel ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 − 𝑌) = 𝑍 ↔ (𝑌 + 𝑍) = 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ablgrp 18198 | . . 3 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
2 | ablsubadd.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
3 | ablsubadd.p | . . . 4 ⊢ + = (+g‘𝐺) | |
4 | ablsubadd.m | . . . 4 ⊢ − = (-g‘𝐺) | |
5 | 2, 3, 4 | grpsubadd 17503 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 − 𝑌) = 𝑍 ↔ (𝑍 + 𝑌) = 𝑋)) |
6 | 1, 5 | sylan 488 | . 2 ⊢ ((𝐺 ∈ Abel ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 − 𝑌) = 𝑍 ↔ (𝑍 + 𝑌) = 𝑋)) |
7 | 2, 3 | ablcom 18210 | . . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 + 𝑍) = (𝑍 + 𝑌)) |
8 | 7 | 3adant3r1 1274 | . . 3 ⊢ ((𝐺 ∈ Abel ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑌 + 𝑍) = (𝑍 + 𝑌)) |
9 | 8 | eqeq1d 2624 | . 2 ⊢ ((𝐺 ∈ Abel ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑌 + 𝑍) = 𝑋 ↔ (𝑍 + 𝑌) = 𝑋)) |
10 | 6, 9 | bitr4d 271 | 1 ⊢ ((𝐺 ∈ Abel ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 − 𝑌) = 𝑍 ↔ (𝑌 + 𝑍) = 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 Grpcgrp 17422 -gcsg 17424 Abelcabl 18194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-cmn 18195 df-abl 18196 |
This theorem is referenced by: lmodvsubadd 18914 |
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