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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > grpoeqdivid | Structured version Visualization version GIF version |
Description: Two group elements are equal iff their quotient is the identity. (Contributed by Jeff Madsen, 6-Jan-2011.) |
Ref | Expression |
---|---|
grpeqdivid.1 | ⊢ 𝑋 = ran 𝐺 |
grpeqdivid.2 | ⊢ 𝑈 = (GId‘𝐺) |
grpeqdivid.3 | ⊢ 𝐷 = ( /𝑔 ‘𝐺) |
Ref | Expression |
---|---|
grpoeqdivid | ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 = 𝐵 ↔ (𝐴𝐷𝐵) = 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpeqdivid.1 | . . . . 5 ⊢ 𝑋 = ran 𝐺 | |
2 | grpeqdivid.3 | . . . . 5 ⊢ 𝐷 = ( /𝑔 ‘𝐺) | |
3 | grpeqdivid.2 | . . . . 5 ⊢ 𝑈 = (GId‘𝐺) | |
4 | 1, 2, 3 | grpodivid 28010 | . . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝐵𝐷𝐵) = 𝑈) |
5 | 4 | 3adant2 1124 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝐷𝐵) = 𝑈) |
6 | oveq1 7023 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴𝐷𝐵) = (𝐵𝐷𝐵)) | |
7 | 6 | eqeq1d 2797 | . . 3 ⊢ (𝐴 = 𝐵 → ((𝐴𝐷𝐵) = 𝑈 ↔ (𝐵𝐷𝐵) = 𝑈)) |
8 | 5, 7 | syl5ibrcom 248 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 = 𝐵 → (𝐴𝐷𝐵) = 𝑈)) |
9 | oveq1 7023 | . . 3 ⊢ ((𝐴𝐷𝐵) = 𝑈 → ((𝐴𝐷𝐵)𝐺𝐵) = (𝑈𝐺𝐵)) | |
10 | 1, 2 | grponpcan 28011 | . . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴) |
11 | 1, 3 | grpolid 27984 | . . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝑈𝐺𝐵) = 𝐵) |
12 | 11 | 3adant2 1124 | . . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑈𝐺𝐵) = 𝐵) |
13 | 10, 12 | eqeq12d 2810 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝐴𝐷𝐵)𝐺𝐵) = (𝑈𝐺𝐵) ↔ 𝐴 = 𝐵)) |
14 | 9, 13 | syl5ib 245 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵) = 𝑈 → 𝐴 = 𝐵)) |
15 | 8, 14 | impbid 213 | 1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 = 𝐵 ↔ (𝐴𝐷𝐵) = 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ w3a 1080 = wceq 1522 ∈ wcel 2081 ran crn 5444 ‘cfv 6225 (class class class)co 7016 GrpOpcgr 27957 GIdcgi 27958 /𝑔 cgs 27960 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-1st 7545 df-2nd 7546 df-grpo 27961 df-gid 27962 df-ginv 27963 df-gdiv 27964 |
This theorem is referenced by: grpokerinj 34703 dmncan1 34886 |
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