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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cntzsnid | Structured version Visualization version GIF version |
Description: The centralizer of the identity element is the whole base set. (Contributed by Thierry Arnoux, 27-Nov-2023.) |
Ref | Expression |
---|---|
cntzun.b | ⊢ 𝐵 = (Base‘𝑀) |
cntzun.z | ⊢ 𝑍 = (Cntz‘𝑀) |
cntzsnid.1 | ⊢ 0 = (0g‘𝑀) |
Ref | Expression |
---|---|
cntzsnid | ⊢ (𝑀 ∈ Mnd → (𝑍‘{ 0 }) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cntzun.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑀) | |
2 | cntzsnid.1 | . . . . 5 ⊢ 0 = (0g‘𝑀) | |
3 | 1, 2 | mndidcl 18775 | . . . 4 ⊢ (𝑀 ∈ Mnd → 0 ∈ 𝐵) |
4 | eqid 2735 | . . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
5 | cntzun.z | . . . . 5 ⊢ 𝑍 = (Cntz‘𝑀) | |
6 | 1, 4, 5 | elcntzsn 19356 | . . . 4 ⊢ ( 0 ∈ 𝐵 → (𝑥 ∈ (𝑍‘{ 0 }) ↔ (𝑥 ∈ 𝐵 ∧ (𝑥(+g‘𝑀) 0 ) = ( 0 (+g‘𝑀)𝑥)))) |
7 | 3, 6 | syl 17 | . . 3 ⊢ (𝑀 ∈ Mnd → (𝑥 ∈ (𝑍‘{ 0 }) ↔ (𝑥 ∈ 𝐵 ∧ (𝑥(+g‘𝑀) 0 ) = ( 0 (+g‘𝑀)𝑥)))) |
8 | 1, 4, 2 | mndrid 18781 | . . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → (𝑥(+g‘𝑀) 0 ) = 𝑥) |
9 | 1, 4, 2 | mndlid 18780 | . . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → ( 0 (+g‘𝑀)𝑥) = 𝑥) |
10 | 8, 9 | eqtr4d 2778 | . . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → (𝑥(+g‘𝑀) 0 ) = ( 0 (+g‘𝑀)𝑥)) |
11 | 10 | ex 412 | . . . 4 ⊢ (𝑀 ∈ Mnd → (𝑥 ∈ 𝐵 → (𝑥(+g‘𝑀) 0 ) = ( 0 (+g‘𝑀)𝑥))) |
12 | 11 | pm4.71d 561 | . . 3 ⊢ (𝑀 ∈ Mnd → (𝑥 ∈ 𝐵 ↔ (𝑥 ∈ 𝐵 ∧ (𝑥(+g‘𝑀) 0 ) = ( 0 (+g‘𝑀)𝑥)))) |
13 | 7, 12 | bitr4d 282 | . 2 ⊢ (𝑀 ∈ Mnd → (𝑥 ∈ (𝑍‘{ 0 }) ↔ 𝑥 ∈ 𝐵)) |
14 | 13 | eqrdv 2733 | 1 ⊢ (𝑀 ∈ Mnd → (𝑍‘{ 0 }) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {csn 4631 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 0gc0g 17486 Mndcmnd 18760 Cntzccntz 19346 |
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-rmo 3378 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-riota 7388 df-ov 7434 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-cntz 19348 |
This theorem is referenced by: (None) |
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