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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 3499 | . 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V) | |
2 | p0val.z | . . 3 ⊢ 0 = (0.‘𝐾) | |
3 | fveq2 6907 | . . . . . 6 ⊢ (𝑝 = 𝐾 → (glb‘𝑝) = (glb‘𝐾)) | |
4 | p0val.g | . . . . . 6 ⊢ 𝐺 = (glb‘𝐾) | |
5 | 3, 4 | eqtr4di 2793 | . . . . 5 ⊢ (𝑝 = 𝐾 → (glb‘𝑝) = 𝐺) |
6 | fveq2 6907 | . . . . . 6 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾)) | |
7 | p0val.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
8 | 6, 7 | eqtr4di 2793 | . . . . 5 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵) |
9 | 5, 8 | fveq12d 6914 | . . . 4 ⊢ (𝑝 = 𝐾 → ((glb‘𝑝)‘(Base‘𝑝)) = (𝐺‘𝐵)) |
10 | df-p0 18483 | . . . 4 ⊢ 0. = (𝑝 ∈ V ↦ ((glb‘𝑝)‘(Base‘𝑝))) | |
11 | fvex 6920 | . . . 4 ⊢ (𝐺‘𝐵) ∈ V | |
12 | 9, 10, 11 | fvmpt 7016 | . . 3 ⊢ (𝐾 ∈ V → (0.‘𝐾) = (𝐺‘𝐵)) |
13 | 2, 12 | eqtrid 2787 | . 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 1537 ∈ wcel 2106 Vcvv 3478 ‘cfv 6563 Basecbs 17245 glbcglb 18368 0.cp0 18481 |
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-p0 18483 |
This theorem is referenced by: p0le 18487 clatp0cl 32951 xrsp0 32997 op0cl 39166 atl0cl 39285 |
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