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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 6103 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (Cntz‘𝑚) = (Cntz‘𝑀)) | |
2 | cntrval.z | . . . . . 6 ⊢ 𝑍 = (Cntz‘𝑀) | |
3 | 1, 2 | syl6eqr 2662 | . . . . 5 ⊢ (𝑚 = 𝑀 → (Cntz‘𝑚) = 𝑍) |
4 | fveq2 6103 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀)) | |
5 | cntrval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑀) | |
6 | 4, 5 | syl6eqr 2662 | . . . . 5 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵) |
7 | 3, 6 | fveq12d 6109 | . . . 4 ⊢ (𝑚 = 𝑀 → ((Cntz‘𝑚)‘(Base‘𝑚)) = (𝑍‘𝐵)) |
8 | df-cntr 17574 | . . . 4 ⊢ Cntr = (𝑚 ∈ V ↦ ((Cntz‘𝑚)‘(Base‘𝑚))) | |
9 | fvex 6113 | . . . 4 ⊢ (𝑍‘𝐵) ∈ V | |
10 | 7, 8, 9 | fvmpt 6191 | . . 3 ⊢ (𝑀 ∈ V → (Cntr‘𝑀) = (𝑍‘𝐵)) |
11 | 10 | eqcomd 2616 | . 2 ⊢ (𝑀 ∈ V → (𝑍‘𝐵) = (Cntr‘𝑀)) |
12 | 0fv 6137 | . . 3 ⊢ (∅‘𝐵) = ∅ | |
13 | fvprc 6097 | . . . . 5 ⊢ (¬ 𝑀 ∈ V → (Cntz‘𝑀) = ∅) | |
14 | 2, 13 | syl5eq 2656 | . . . 4 ⊢ (¬ 𝑀 ∈ V → 𝑍 = ∅) |
15 | 14 | fveq1d 6105 | . . 3 ⊢ (¬ 𝑀 ∈ V → (𝑍‘𝐵) = (∅‘𝐵)) |
16 | fvprc 6097 | . . 3 ⊢ (¬ 𝑀 ∈ V → (Cntr‘𝑀) = ∅) | |
17 | 12, 15, 16 | 3eqtr4a 2670 | . 2 ⊢ (¬ 𝑀 ∈ V → (𝑍‘𝐵) = (Cntr‘𝑀)) |
18 | 11, 17 | pm2.61i 175 | 1 ⊢ (𝑍‘𝐵) = (Cntr‘𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 ‘cfv 5804 Basecbs 15695 Cntzccntz 17571 Cntrccntr 17572 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-cntr 17574 |
This theorem is referenced by: cntri 17586 cntrsubgnsg 17596 cntrnsg 17597 oppgcntr 17618 |
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