![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 3493 | . 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V) | |
2 | p1val.t | . . 3 ⊢ 1 = (1.‘𝐾) | |
3 | fveq2 6888 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (lub‘𝑘) = (lub‘𝐾)) | |
4 | p1val.u | . . . . . 6 ⊢ 𝑈 = (lub‘𝐾) | |
5 | 3, 4 | eqtr4di 2791 | . . . . 5 ⊢ (𝑘 = 𝐾 → (lub‘𝑘) = 𝑈) |
6 | fveq2 6888 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾)) | |
7 | p1val.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
8 | 6, 7 | eqtr4di 2791 | . . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵) |
9 | 5, 8 | fveq12d 6895 | . . . 4 ⊢ (𝑘 = 𝐾 → ((lub‘𝑘)‘(Base‘𝑘)) = (𝑈‘𝐵)) |
10 | df-p1 18375 | . . . 4 ⊢ 1. = (𝑘 ∈ V ↦ ((lub‘𝑘)‘(Base‘𝑘))) | |
11 | fvex 6901 | . . . 4 ⊢ (𝑈‘𝐵) ∈ V | |
12 | 9, 10, 11 | fvmpt 6994 | . . 3 ⊢ (𝐾 ∈ V → (1.‘𝐾) = (𝑈‘𝐵)) |
13 | 2, 12 | eqtrid 2785 | . 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 1542 ∈ wcel 2107 Vcvv 3475 ‘cfv 6540 Basecbs 17140 lubclub 18258 1.cp1 18373 |
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 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6492 df-fun 6542 df-fv 6548 df-p1 18375 |
This theorem is referenced by: ple1 18379 clatp1cl 32125 xrsp1 32161 op1cl 37993 |
Copyright terms: Public domain | W3C validator |