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 17922 | . . . 4 ⊢ (𝑀 ∈ Mnd → 0 ∈ 𝐵) |
4 | eqid 2820 | . . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
5 | cntzun.z | . . . . 5 ⊢ 𝑍 = (Cntz‘𝑀) | |
6 | 1, 4, 5 | elcntzsn 18451 | . . . 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 17928 | . . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → (𝑥(+g‘𝑀) 0 ) = 𝑥) |
9 | 1, 4, 2 | mndlid 17927 | . . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → ( 0 (+g‘𝑀)𝑥) = 𝑥) |
10 | 8, 9 | eqtr4d 2858 | . . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → (𝑥(+g‘𝑀) 0 ) = ( 0 (+g‘𝑀)𝑥)) |
11 | 10 | ex 415 | . . . 4 ⊢ (𝑀 ∈ Mnd → (𝑥 ∈ 𝐵 → (𝑥(+g‘𝑀) 0 ) = ( 0 (+g‘𝑀)𝑥))) |
12 | 11 | pm4.71d 564 | . . 3 ⊢ (𝑀 ∈ Mnd → (𝑥 ∈ 𝐵 ↔ (𝑥 ∈ 𝐵 ∧ (𝑥(+g‘𝑀) 0 ) = ( 0 (+g‘𝑀)𝑥)))) |
13 | 7, 12 | bitr4d 284 | . 2 ⊢ (𝑀 ∈ Mnd → (𝑥 ∈ (𝑍‘{ 0 }) ↔ 𝑥 ∈ 𝐵)) |
14 | 13 | eqrdv 2818 | 1 ⊢ (𝑀 ∈ Mnd → (𝑍‘{ 0 }) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 {csn 4564 ‘cfv 6352 (class class class)co 7153 Basecbs 16479 +gcplusg 16561 0gc0g 16709 Mndcmnd 17907 Cntzccntz 18441 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5187 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3495 df-sbc 3771 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4836 df-iun 4918 df-br 5064 df-opab 5126 df-mpt 5144 df-id 5457 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-riota 7111 df-ov 7156 df-0g 16711 df-mgm 17848 df-sgrp 17897 df-mnd 17908 df-cntz 18443 |
This theorem is referenced by: (None) |
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