![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cntrval | Structured version Visualization version GIF version |
Description: Substitute definition of the center. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
Ref | Expression |
---|---|
cntrval.b | ⊢ 𝐵 = (Base‘𝑀) |
cntrval.z | ⊢ 𝑍 = (Cntz‘𝑀) |
Ref | Expression |
---|---|
cntrval | ⊢ (𝑍‘𝐵) = (Cntr‘𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6645 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (Cntz‘𝑚) = (Cntz‘𝑀)) | |
2 | cntrval.z | . . . . . 6 ⊢ 𝑍 = (Cntz‘𝑀) | |
3 | 1, 2 | eqtr4di 2851 | . . . . 5 ⊢ (𝑚 = 𝑀 → (Cntz‘𝑚) = 𝑍) |
4 | fveq2 6645 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀)) | |
5 | cntrval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑀) | |
6 | 4, 5 | eqtr4di 2851 | . . . . 5 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵) |
7 | 3, 6 | fveq12d 6652 | . . . 4 ⊢ (𝑚 = 𝑀 → ((Cntz‘𝑚)‘(Base‘𝑚)) = (𝑍‘𝐵)) |
8 | df-cntr 18440 | . . . 4 ⊢ Cntr = (𝑚 ∈ V ↦ ((Cntz‘𝑚)‘(Base‘𝑚))) | |
9 | fvex 6658 | . . . 4 ⊢ (𝑍‘𝐵) ∈ V | |
10 | 7, 8, 9 | fvmpt 6745 | . . 3 ⊢ (𝑀 ∈ V → (Cntr‘𝑀) = (𝑍‘𝐵)) |
11 | 10 | eqcomd 2804 | . 2 ⊢ (𝑀 ∈ V → (𝑍‘𝐵) = (Cntr‘𝑀)) |
12 | 0fv 6684 | . . 3 ⊢ (∅‘𝐵) = ∅ | |
13 | fvprc 6638 | . . . . 5 ⊢ (¬ 𝑀 ∈ V → (Cntz‘𝑀) = ∅) | |
14 | 2, 13 | syl5eq 2845 | . . . 4 ⊢ (¬ 𝑀 ∈ V → 𝑍 = ∅) |
15 | 14 | fveq1d 6647 | . . 3 ⊢ (¬ 𝑀 ∈ V → (𝑍‘𝐵) = (∅‘𝐵)) |
16 | fvprc 6638 | . . 3 ⊢ (¬ 𝑀 ∈ V → (Cntr‘𝑀) = ∅) | |
17 | 12, 15, 16 | 3eqtr4a 2859 | . 2 ⊢ (¬ 𝑀 ∈ V → (𝑍‘𝐵) = (Cntr‘𝑀)) |
18 | 11, 17 | pm2.61i 185 | 1 ⊢ (𝑍‘𝐵) = (Cntr‘𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 ‘cfv 6324 Basecbs 16475 Cntzccntz 18437 Cntrccntr 18438 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-cntr 18440 |
This theorem is referenced by: cntrss 18452 cntri 18453 cntrsubgnsg 18463 cntrnsg 18464 oppgcntr 18485 cntrcmnd 18955 cntrabl 18956 primefld 19577 cntrcrng 30747 |
Copyright terms: Public domain | W3C validator |