![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2731 | . . . 4 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺) | |
4 | 2, 3 | cntrval 19231 | . . 3 ⊢ ((Cntz‘𝐺)‘𝐵) = (Cntr‘𝐺) |
5 | ssid 4004 | . . . 4 ⊢ 𝐵 ⊆ 𝐵 | |
6 | eqid 2731 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
7 | 2, 6, 3 | cntzval 19233 | . . . 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 2765 | . 2 ⊢ 𝑍 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)} |
10 | 2, 6 | cmncom 19714 | . . . . 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 3145 | . . 3 ⊢ ((𝐺 ∈ CMnd ∧ 𝑥 ∈ 𝐵) → ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)) |
13 | 12 | rabeqcda 3442 | . 2 ⊢ (𝐺 ∈ CMnd → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)} = 𝐵) |
14 | 9, 13 | eqtr2id 2784 | 1 ⊢ (𝐺 ∈ CMnd → 𝐵 = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∀wral 3060 {crab 3431 ⊆ wss 3948 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 +gcplusg 17204 Cntzccntz 19227 Cntrccntr 19228 CMndccmn 19696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-cntz 19229 df-cntr 19230 df-cmn 19698 |
This theorem is referenced by: crngbascntr 20156 |
Copyright terms: Public domain | W3C validator |