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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 18762 | . . . 4 ⊢ (𝑀 ∈ Mnd → 0 ∈ 𝐵) | 
| 4 | eqid 2737 | . . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
| 5 | cntzun.z | . . . . 5 ⊢ 𝑍 = (Cntz‘𝑀) | |
| 6 | 1, 4, 5 | elcntzsn 19343 | . . . 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 18768 | . . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → (𝑥(+g‘𝑀) 0 ) = 𝑥) | 
| 9 | 1, 4, 2 | mndlid 18767 | . . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → ( 0 (+g‘𝑀)𝑥) = 𝑥) | 
| 10 | 8, 9 | eqtr4d 2780 | . . . . 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 2735 | 1 ⊢ (𝑀 ∈ Mnd → (𝑍‘{ 0 }) = 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {csn 4626 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 0gc0g 17484 Mndcmnd 18747 Cntzccntz 19333 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-cntz 19335 | 
| This theorem is referenced by: (None) | 
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