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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 6907 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (Cntz‘𝑚) = (Cntz‘𝑀)) | |
2 | cntrval.z | . . . . . 6 ⊢ 𝑍 = (Cntz‘𝑀) | |
3 | 1, 2 | eqtr4di 2793 | . . . . 5 ⊢ (𝑚 = 𝑀 → (Cntz‘𝑚) = 𝑍) |
4 | fveq2 6907 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀)) | |
5 | cntrval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑀) | |
6 | 4, 5 | eqtr4di 2793 | . . . . 5 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵) |
7 | 3, 6 | fveq12d 6914 | . . . 4 ⊢ (𝑚 = 𝑀 → ((Cntz‘𝑚)‘(Base‘𝑚)) = (𝑍‘𝐵)) |
8 | df-cntr 19349 | . . . 4 ⊢ Cntr = (𝑚 ∈ V ↦ ((Cntz‘𝑚)‘(Base‘𝑚))) | |
9 | fvex 6920 | . . . 4 ⊢ (𝑍‘𝐵) ∈ V | |
10 | 7, 8, 9 | fvmpt 7016 | . . 3 ⊢ (𝑀 ∈ V → (Cntr‘𝑀) = (𝑍‘𝐵)) |
11 | 10 | eqcomd 2741 | . 2 ⊢ (𝑀 ∈ V → (𝑍‘𝐵) = (Cntr‘𝑀)) |
12 | 0fv 6951 | . . 3 ⊢ (∅‘𝐵) = ∅ | |
13 | fvprc 6899 | . . . . 5 ⊢ (¬ 𝑀 ∈ V → (Cntz‘𝑀) = ∅) | |
14 | 2, 13 | eqtrid 2787 | . . . 4 ⊢ (¬ 𝑀 ∈ V → 𝑍 = ∅) |
15 | 14 | fveq1d 6909 | . . 3 ⊢ (¬ 𝑀 ∈ V → (𝑍‘𝐵) = (∅‘𝐵)) |
16 | fvprc 6899 | . . 3 ⊢ (¬ 𝑀 ∈ V → (Cntr‘𝑀) = ∅) | |
17 | 12, 15, 16 | 3eqtr4a 2801 | . 2 ⊢ (¬ 𝑀 ∈ V → (𝑍‘𝐵) = (Cntr‘𝑀)) |
18 | 11, 17 | pm2.61i 182 | 1 ⊢ (𝑍‘𝐵) = (Cntr‘𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 ‘cfv 6563 Basecbs 17245 Cntzccntz 19346 Cntrccntr 19347 |
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-sep 5302 ax-nul 5312 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-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 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-iota 6516 df-fun 6565 df-fv 6571 df-cntr 19349 |
This theorem is referenced by: elcntr 19361 cntrss 19362 cntri 19363 cntrsubgnsg 19374 cntrnsg 19375 oppgcntr 19399 cmnbascntr 19838 cntrcmnd 19875 cntrabl 19876 primefld 20823 rng2idl1cntr 21333 cntrcrng 33056 |
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