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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 3466 | . 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V) | |
2 | p0val.z | . . 3 ⊢ 0 = (0.‘𝐾) | |
3 | fveq2 6847 | . . . . . 6 ⊢ (𝑝 = 𝐾 → (glb‘𝑝) = (glb‘𝐾)) | |
4 | p0val.g | . . . . . 6 ⊢ 𝐺 = (glb‘𝐾) | |
5 | 3, 4 | eqtr4di 2795 | . . . . 5 ⊢ (𝑝 = 𝐾 → (glb‘𝑝) = 𝐺) |
6 | fveq2 6847 | . . . . . 6 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾)) | |
7 | p0val.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
8 | 6, 7 | eqtr4di 2795 | . . . . 5 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵) |
9 | 5, 8 | fveq12d 6854 | . . . 4 ⊢ (𝑝 = 𝐾 → ((glb‘𝑝)‘(Base‘𝑝)) = (𝐺‘𝐵)) |
10 | df-p0 18321 | . . . 4 ⊢ 0. = (𝑝 ∈ V ↦ ((glb‘𝑝)‘(Base‘𝑝))) | |
11 | fvex 6860 | . . . 4 ⊢ (𝐺‘𝐵) ∈ V | |
12 | 9, 10, 11 | fvmpt 6953 | . . 3 ⊢ (𝐾 ∈ V → (0.‘𝐾) = (𝐺‘𝐵)) |
13 | 2, 12 | eqtrid 2789 | . 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 1542 ∈ wcel 2107 Vcvv 3448 ‘cfv 6501 Basecbs 17090 glbcglb 18206 0.cp0 18319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-iota 6453 df-fun 6503 df-fv 6509 df-p0 18321 |
This theorem is referenced by: p0le 18325 clatp0cl 31878 xrsp0 31914 op0cl 37675 atl0cl 37794 |
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