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Mirrors > Home > MPE Home > Th. List > cntrss | Structured version Visualization version GIF version |
Description: The center is a subset of the base field. (Contributed by Thierry Arnoux, 21-Aug-2023.) |
Ref | Expression |
---|---|
cntrss.1 | ⊢ 𝐵 = (Base‘𝑀) |
Ref | Expression |
---|---|
cntrss | ⊢ (Cntr‘𝑀) ⊆ 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cntrss.1 | . . 3 ⊢ 𝐵 = (Base‘𝑀) | |
2 | eqid 2740 | . . 3 ⊢ (Cntz‘𝑀) = (Cntz‘𝑀) | |
3 | 1, 2 | cntrval 18915 | . 2 ⊢ ((Cntz‘𝑀)‘𝐵) = (Cntr‘𝑀) |
4 | 1, 2 | cntzssv 18924 | . 2 ⊢ ((Cntz‘𝑀)‘𝐵) ⊆ 𝐵 |
5 | 3, 4 | eqsstrri 3961 | 1 ⊢ (Cntr‘𝑀) ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ⊆ wss 3892 ‘cfv 6431 Basecbs 16902 Cntzccntz 18911 Cntrccntr 18912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-ov 7272 df-cntz 18913 df-cntr 18914 |
This theorem is referenced by: cntrcmnd 19433 primefld 20063 |
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