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Mirrors > Home > MPE Home > Th. List > cntzcmnss | Structured version Visualization version GIF version |
Description: Any subset in a commutative monoid is a subset of its centralizer. (Contributed by AV, 12-Jan-2019.) |
Ref | Expression |
---|---|
cntzcmnss.b | ⊢ 𝐵 = (Base‘𝐺) |
cntzcmnss.z | ⊢ 𝑍 = (Cntz‘𝐺) |
Ref | Expression |
---|---|
cntzcmnss | ⊢ ((𝐺 ∈ CMnd ∧ 𝑆 ⊆ 𝐵) → 𝑆 ⊆ (𝑍‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cntzcmnss.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | cntzcmnss.z | . . 3 ⊢ 𝑍 = (Cntz‘𝐺) | |
3 | 1, 2 | cntzcmn 18959 | . 2 ⊢ ((𝐺 ∈ CMnd ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) = 𝐵) |
4 | sseq2 3992 | . . . . 5 ⊢ (𝐵 = (𝑍‘𝑆) → (𝑆 ⊆ 𝐵 ↔ 𝑆 ⊆ (𝑍‘𝑆))) | |
5 | 4 | eqcoms 2829 | . . . 4 ⊢ ((𝑍‘𝑆) = 𝐵 → (𝑆 ⊆ 𝐵 ↔ 𝑆 ⊆ (𝑍‘𝑆))) |
6 | 5 | biimpd 231 | . . 3 ⊢ ((𝑍‘𝑆) = 𝐵 → (𝑆 ⊆ 𝐵 → 𝑆 ⊆ (𝑍‘𝑆))) |
7 | 6 | adantld 493 | . 2 ⊢ ((𝑍‘𝑆) = 𝐵 → ((𝐺 ∈ CMnd ∧ 𝑆 ⊆ 𝐵) → 𝑆 ⊆ (𝑍‘𝑆))) |
8 | 3, 7 | mpcom 38 | 1 ⊢ ((𝐺 ∈ CMnd ∧ 𝑆 ⊆ 𝐵) → 𝑆 ⊆ (𝑍‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ‘cfv 6354 Basecbs 16482 Cntzccntz 18444 CMndccmn 18905 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-cntz 18446 df-cmn 18907 |
This theorem is referenced by: smadiadetlem3lem2 21275 |
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