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Mirrors > Home > MPE Home > Th. List > p1val | Structured version Visualization version GIF version |
Description: Value of poset zero. (Contributed by NM, 22-Oct-2011.) |
Ref | Expression |
---|---|
p1val.b | ⊢ 𝐵 = (Base‘𝐾) |
p1val.u | ⊢ 𝑈 = (lub‘𝐾) |
p1val.t | ⊢ 1 = (1.‘𝐾) |
Ref | Expression |
---|---|
p1val | ⊢ (𝐾 ∈ 𝑉 → 1 = (𝑈‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3515 | . 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V) | |
2 | p1val.t | . . 3 ⊢ 1 = (1.‘𝐾) | |
3 | fveq2 6673 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (lub‘𝑘) = (lub‘𝐾)) | |
4 | p1val.u | . . . . . 6 ⊢ 𝑈 = (lub‘𝐾) | |
5 | 3, 4 | syl6eqr 2877 | . . . . 5 ⊢ (𝑘 = 𝐾 → (lub‘𝑘) = 𝑈) |
6 | fveq2 6673 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾)) | |
7 | p1val.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
8 | 6, 7 | syl6eqr 2877 | . . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵) |
9 | 5, 8 | fveq12d 6680 | . . . 4 ⊢ (𝑘 = 𝐾 → ((lub‘𝑘)‘(Base‘𝑘)) = (𝑈‘𝐵)) |
10 | df-p1 17653 | . . . 4 ⊢ 1. = (𝑘 ∈ V ↦ ((lub‘𝑘)‘(Base‘𝑘))) | |
11 | fvex 6686 | . . . 4 ⊢ (𝑈‘𝐵) ∈ V | |
12 | 9, 10, 11 | fvmpt 6771 | . . 3 ⊢ (𝐾 ∈ V → (1.‘𝐾) = (𝑈‘𝐵)) |
13 | 2, 12 | syl5eq 2871 | . 2 ⊢ (𝐾 ∈ V → 1 = (𝑈‘𝐵)) |
14 | 1, 13 | syl 17 | 1 ⊢ (𝐾 ∈ 𝑉 → 1 = (𝑈‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ‘cfv 6358 Basecbs 16486 lubclub 17555 1.cp1 17651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-iota 6317 df-fun 6360 df-fv 6366 df-p1 17653 |
This theorem is referenced by: ple1 17657 clatp1cl 30663 xrsp1 30673 op1cl 36325 |
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