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Mirrors > Home > MPE Home > Th. List > p0val | Structured version Visualization version GIF version |
Description: Value of poset zero. (Contributed by NM, 12-Oct-2011.) |
Ref | Expression |
---|---|
p0val.b | ⊢ 𝐵 = (Base‘𝐾) |
p0val.g | ⊢ 𝐺 = (glb‘𝐾) |
p0val.z | ⊢ 0 = (0.‘𝐾) |
Ref | Expression |
---|---|
p0val | ⊢ (𝐾 ∈ 𝑉 → 0 = (𝐺‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3490 | . 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V) | |
2 | p0val.z | . . 3 ⊢ 0 = (0.‘𝐾) | |
3 | fveq2 6897 | . . . . . 6 ⊢ (𝑝 = 𝐾 → (glb‘𝑝) = (glb‘𝐾)) | |
4 | p0val.g | . . . . . 6 ⊢ 𝐺 = (glb‘𝐾) | |
5 | 3, 4 | eqtr4di 2786 | . . . . 5 ⊢ (𝑝 = 𝐾 → (glb‘𝑝) = 𝐺) |
6 | fveq2 6897 | . . . . . 6 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾)) | |
7 | p0val.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
8 | 6, 7 | eqtr4di 2786 | . . . . 5 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵) |
9 | 5, 8 | fveq12d 6904 | . . . 4 ⊢ (𝑝 = 𝐾 → ((glb‘𝑝)‘(Base‘𝑝)) = (𝐺‘𝐵)) |
10 | df-p0 18416 | . . . 4 ⊢ 0. = (𝑝 ∈ V ↦ ((glb‘𝑝)‘(Base‘𝑝))) | |
11 | fvex 6910 | . . . 4 ⊢ (𝐺‘𝐵) ∈ V | |
12 | 9, 10, 11 | fvmpt 7005 | . . 3 ⊢ (𝐾 ∈ V → (0.‘𝐾) = (𝐺‘𝐵)) |
13 | 2, 12 | eqtrid 2780 | . 2 ⊢ (𝐾 ∈ V → 0 = (𝐺‘𝐵)) |
14 | 1, 13 | syl 17 | 1 ⊢ (𝐾 ∈ 𝑉 → 0 = (𝐺‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ‘cfv 6548 Basecbs 17179 glbcglb 18301 0.cp0 18414 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6500 df-fun 6550 df-fv 6556 df-p0 18416 |
This theorem is referenced by: p0le 18420 clatp0cl 32703 xrsp0 32739 op0cl 38656 atl0cl 38775 |
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