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Mirrors > Home > MPE Home > Th. List > cmnbascntr | Structured version Visualization version GIF version |
Description: The base set of a commutative monoid is its center. (Contributed by SN, 21-Mar-2025.) |
Ref | Expression |
---|---|
cmnbascntr.b | ⊢ 𝐵 = (Base‘𝐺) |
cmnbascntr.z | ⊢ 𝑍 = (Cntr‘𝐺) |
Ref | Expression |
---|---|
cmnbascntr | ⊢ (𝐺 ∈ CMnd → 𝐵 = 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cmnbascntr.z | . . 3 ⊢ 𝑍 = (Cntr‘𝐺) | |
2 | cmnbascntr.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
3 | eqid 2735 | . . . 4 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺) | |
4 | 2, 3 | cntrval 19350 | . . 3 ⊢ ((Cntz‘𝐺)‘𝐵) = (Cntr‘𝐺) |
5 | ssid 4018 | . . . 4 ⊢ 𝐵 ⊆ 𝐵 | |
6 | eqid 2735 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
7 | 2, 6, 3 | cntzval 19352 | . . . 4 ⊢ (𝐵 ⊆ 𝐵 → ((Cntz‘𝐺)‘𝐵) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)}) |
8 | 5, 7 | ax-mp 5 | . . 3 ⊢ ((Cntz‘𝐺)‘𝐵) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)} |
9 | 1, 4, 8 | 3eqtr2i 2769 | . 2 ⊢ 𝑍 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)} |
10 | 2, 6 | cmncom 19831 | . . . . 5 ⊢ ((𝐺 ∈ CMnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)) |
11 | 10 | 3expa 1117 | . . . 4 ⊢ (((𝐺 ∈ CMnd ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)) |
12 | 11 | ralrimiva 3144 | . . 3 ⊢ ((𝐺 ∈ CMnd ∧ 𝑥 ∈ 𝐵) → ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)) |
13 | 12 | rabeqcda 3445 | . 2 ⊢ (𝐺 ∈ CMnd → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)} = 𝐵) |
14 | 9, 13 | eqtr2id 2788 | 1 ⊢ (𝐺 ∈ CMnd → 𝐵 = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 {crab 3433 ⊆ wss 3963 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 Cntzccntz 19346 Cntrccntr 19347 CMndccmn 19813 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-cntz 19348 df-cntr 19349 df-cmn 19815 |
This theorem is referenced by: crngbascntr 20274 |
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