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Mirrors > Home > MPE Home > Th. List > cntzel | Structured version Visualization version GIF version |
Description: Membership in a centralizer. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
Ref | Expression |
---|---|
cntzfval.b | ⊢ 𝐵 = (Base‘𝑀) |
cntzfval.p | ⊢ + = (+g‘𝑀) |
cntzfval.z | ⊢ 𝑍 = (Cntz‘𝑀) |
Ref | Expression |
---|---|
cntzel | ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ (𝑍‘𝑆) ↔ ∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cntzfval.b | . . 3 ⊢ 𝐵 = (Base‘𝑀) | |
2 | cntzfval.p | . . 3 ⊢ + = (+g‘𝑀) | |
3 | cntzfval.z | . . 3 ⊢ 𝑍 = (Cntz‘𝑀) | |
4 | 1, 2, 3 | elcntz 18573 | . 2 ⊢ (𝑆 ⊆ 𝐵 → (𝑋 ∈ (𝑍‘𝑆) ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋)))) |
5 | 4 | baibd 543 | 1 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ (𝑍‘𝑆) ↔ ∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∀wral 3054 ⊆ wss 3844 ‘cfv 6340 (class class class)co 7173 Basecbs 16589 +gcplusg 16671 Cntzccntz 18566 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-id 5430 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-ov 7176 df-cntz 18568 |
This theorem is referenced by: cntzsubg 18588 cntzcmn 19082 cntzsubr 19690 cntzsdrg 19703 |
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