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Mirrors > Home > MPE Home > Th. List > cntzcmn | Structured version Visualization version GIF version |
Description: The centralizer of any subset in a commutative monoid is the whole monoid. (Contributed by Mario Carneiro, 3-Oct-2015.) |
Ref | Expression |
---|---|
cntzcmn.b | ⊢ 𝐵 = (Base‘𝐺) |
cntzcmn.z | ⊢ 𝑍 = (Cntz‘𝐺) |
Ref | Expression |
---|---|
cntzcmn | ⊢ ((𝐺 ∈ CMnd ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cntzcmn.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
2 | cntzcmn.z | . . . 4 ⊢ 𝑍 = (Cntz‘𝐺) | |
3 | 1, 2 | cntzssv 18110 | . . 3 ⊢ (𝑍‘𝑆) ⊆ 𝐵 |
4 | 3 | a1i 11 | . 2 ⊢ ((𝐺 ∈ CMnd ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ⊆ 𝐵) |
5 | simpl1 1248 | . . . . . . 7 ⊢ (((𝐺 ∈ CMnd ∧ 𝑆 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝑆) → 𝐺 ∈ CMnd) | |
6 | simpl3 1252 | . . . . . . 7 ⊢ (((𝐺 ∈ CMnd ∧ 𝑆 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝐵) | |
7 | simp2 1173 | . . . . . . . 8 ⊢ ((𝐺 ∈ CMnd ∧ 𝑆 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐵) → 𝑆 ⊆ 𝐵) | |
8 | 7 | sselda 3826 | . . . . . . 7 ⊢ (((𝐺 ∈ CMnd ∧ 𝑆 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝐵) |
9 | eqid 2824 | . . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
10 | 1, 9 | cmncom 18561 | . . . . . . 7 ⊢ ((𝐺 ∈ CMnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)) |
11 | 5, 6, 8, 10 | syl3anc 1496 | . . . . . 6 ⊢ (((𝐺 ∈ CMnd ∧ 𝑆 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝑆) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)) |
12 | 11 | ralrimiva 3174 | . . . . 5 ⊢ ((𝐺 ∈ CMnd ∧ 𝑆 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐵) → ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)) |
13 | 1, 9, 2 | cntzel 18105 | . . . . . 6 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝑍‘𝑆) ↔ ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
14 | 13 | 3adant1 1166 | . . . . 5 ⊢ ((𝐺 ∈ CMnd ∧ 𝑆 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝑍‘𝑆) ↔ ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
15 | 12, 14 | mpbird 249 | . . . 4 ⊢ ((𝐺 ∈ CMnd ∧ 𝑆 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝑍‘𝑆)) |
16 | 15 | 3expia 1156 | . . 3 ⊢ ((𝐺 ∈ CMnd ∧ 𝑆 ⊆ 𝐵) → (𝑥 ∈ 𝐵 → 𝑥 ∈ (𝑍‘𝑆))) |
17 | 16 | ssrdv 3832 | . 2 ⊢ ((𝐺 ∈ CMnd ∧ 𝑆 ⊆ 𝐵) → 𝐵 ⊆ (𝑍‘𝑆)) |
18 | 4, 17 | eqssd 3843 | 1 ⊢ ((𝐺 ∈ CMnd ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ∀wral 3116 ⊆ wss 3797 ‘cfv 6122 (class class class)co 6904 Basecbs 16221 +gcplusg 16304 Cntzccntz 18097 CMndccmn 18545 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-ral 3121 df-rex 3122 df-reu 3123 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-op 4403 df-uni 4658 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-id 5249 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-ov 6907 df-cntz 18099 df-cmn 18547 |
This theorem is referenced by: cntzcmnss 18598 cntzcmnf 18600 ablcntzd 18612 gsumadd 18675 |
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