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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 3513 | . 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V) | |
2 | p0val.z | . . 3 ⊢ 0 = (0.‘𝐾) | |
3 | fveq2 6665 | . . . . . 6 ⊢ (𝑝 = 𝐾 → (glb‘𝑝) = (glb‘𝐾)) | |
4 | p0val.g | . . . . . 6 ⊢ 𝐺 = (glb‘𝐾) | |
5 | 3, 4 | syl6eqr 2874 | . . . . 5 ⊢ (𝑝 = 𝐾 → (glb‘𝑝) = 𝐺) |
6 | fveq2 6665 | . . . . . 6 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾)) | |
7 | p0val.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
8 | 6, 7 | syl6eqr 2874 | . . . . 5 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵) |
9 | 5, 8 | fveq12d 6672 | . . . 4 ⊢ (𝑝 = 𝐾 → ((glb‘𝑝)‘(Base‘𝑝)) = (𝐺‘𝐵)) |
10 | df-p0 17643 | . . . 4 ⊢ 0. = (𝑝 ∈ V ↦ ((glb‘𝑝)‘(Base‘𝑝))) | |
11 | fvex 6678 | . . . 4 ⊢ (𝐺‘𝐵) ∈ V | |
12 | 9, 10, 11 | fvmpt 6763 | . . 3 ⊢ (𝐾 ∈ V → (0.‘𝐾) = (𝐺‘𝐵)) |
13 | 2, 12 | syl5eq 2868 | . 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 1533 ∈ wcel 2110 Vcvv 3495 ‘cfv 6350 Basecbs 16477 glbcglb 17547 0.cp0 17641 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-iota 6309 df-fun 6352 df-fv 6358 df-p0 17643 |
This theorem is referenced by: p0le 17647 clatp0cl 30653 xrsp0 30663 op0cl 36314 atl0cl 36433 |
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