Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrhval | Structured version Visualization version GIF version |
Description: The value of the embedding from the extended real numbers into a complete lattice. (Contributed by Thierry Arnoux, 19-Feb-2018.) |
Ref | Expression |
---|---|
xrhval.b | ⊢ 𝐵 = ((ℝHom‘𝑅) “ ℝ) |
xrhval.l | ⊢ 𝐿 = (glb‘𝑅) |
xrhval.u | ⊢ 𝑈 = (lub‘𝑅) |
Ref | Expression |
---|---|
xrhval | ⊢ (𝑅 ∈ 𝑉 → (ℝ*Hom‘𝑅) = (𝑥 ∈ ℝ* ↦ if(𝑥 ∈ ℝ, ((ℝHom‘𝑅)‘𝑥), if(𝑥 = +∞, (𝑈‘𝐵), (𝐿‘𝐵))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3185 | . 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
2 | fveq2 6103 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (ℝHom‘𝑟) = (ℝHom‘𝑅)) | |
3 | 2 | fveq1d 6105 | . . . . 5 ⊢ (𝑟 = 𝑅 → ((ℝHom‘𝑟)‘𝑥) = ((ℝHom‘𝑅)‘𝑥)) |
4 | fveq2 6103 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (lub‘𝑟) = (lub‘𝑅)) | |
5 | xrhval.u | . . . . . . . 8 ⊢ 𝑈 = (lub‘𝑅) | |
6 | 4, 5 | syl6eqr 2662 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (lub‘𝑟) = 𝑈) |
7 | 2 | imaeq1d 5384 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((ℝHom‘𝑟) “ ℝ) = ((ℝHom‘𝑅) “ ℝ)) |
8 | xrhval.b | . . . . . . . 8 ⊢ 𝐵 = ((ℝHom‘𝑅) “ ℝ) | |
9 | 7, 8 | syl6eqr 2662 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → ((ℝHom‘𝑟) “ ℝ) = 𝐵) |
10 | 6, 9 | fveq12d 6109 | . . . . . 6 ⊢ (𝑟 = 𝑅 → ((lub‘𝑟)‘((ℝHom‘𝑟) “ ℝ)) = (𝑈‘𝐵)) |
11 | fveq2 6103 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (glb‘𝑟) = (glb‘𝑅)) | |
12 | xrhval.l | . . . . . . . 8 ⊢ 𝐿 = (glb‘𝑅) | |
13 | 11, 12 | syl6eqr 2662 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (glb‘𝑟) = 𝐿) |
14 | 13, 9 | fveq12d 6109 | . . . . . 6 ⊢ (𝑟 = 𝑅 → ((glb‘𝑟)‘((ℝHom‘𝑟) “ ℝ)) = (𝐿‘𝐵)) |
15 | 10, 14 | ifeq12d 4056 | . . . . 5 ⊢ (𝑟 = 𝑅 → if(𝑥 = +∞, ((lub‘𝑟)‘((ℝHom‘𝑟) “ ℝ)), ((glb‘𝑟)‘((ℝHom‘𝑟) “ ℝ))) = if(𝑥 = +∞, (𝑈‘𝐵), (𝐿‘𝐵))) |
16 | 3, 15 | ifeq12d 4056 | . . . 4 ⊢ (𝑟 = 𝑅 → if(𝑥 ∈ ℝ, ((ℝHom‘𝑟)‘𝑥), if(𝑥 = +∞, ((lub‘𝑟)‘((ℝHom‘𝑟) “ ℝ)), ((glb‘𝑟)‘((ℝHom‘𝑟) “ ℝ)))) = if(𝑥 ∈ ℝ, ((ℝHom‘𝑅)‘𝑥), if(𝑥 = +∞, (𝑈‘𝐵), (𝐿‘𝐵)))) |
17 | 16 | mpteq2dv 4673 | . . 3 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ ℝ* ↦ if(𝑥 ∈ ℝ, ((ℝHom‘𝑟)‘𝑥), if(𝑥 = +∞, ((lub‘𝑟)‘((ℝHom‘𝑟) “ ℝ)), ((glb‘𝑟)‘((ℝHom‘𝑟) “ ℝ))))) = (𝑥 ∈ ℝ* ↦ if(𝑥 ∈ ℝ, ((ℝHom‘𝑅)‘𝑥), if(𝑥 = +∞, (𝑈‘𝐵), (𝐿‘𝐵))))) |
18 | df-xrh 29389 | . . 3 ⊢ ℝ*Hom = (𝑟 ∈ V ↦ (𝑥 ∈ ℝ* ↦ if(𝑥 ∈ ℝ, ((ℝHom‘𝑟)‘𝑥), if(𝑥 = +∞, ((lub‘𝑟)‘((ℝHom‘𝑟) “ ℝ)), ((glb‘𝑟)‘((ℝHom‘𝑟) “ ℝ)))))) | |
19 | xrex 11705 | . . . 4 ⊢ ℝ* ∈ V | |
20 | 19 | mptex 6390 | . . 3 ⊢ (𝑥 ∈ ℝ* ↦ if(𝑥 ∈ ℝ, ((ℝHom‘𝑅)‘𝑥), if(𝑥 = +∞, (𝑈‘𝐵), (𝐿‘𝐵)))) ∈ V |
21 | 17, 18, 20 | fvmpt 6191 | . 2 ⊢ (𝑅 ∈ V → (ℝ*Hom‘𝑅) = (𝑥 ∈ ℝ* ↦ if(𝑥 ∈ ℝ, ((ℝHom‘𝑅)‘𝑥), if(𝑥 = +∞, (𝑈‘𝐵), (𝐿‘𝐵))))) |
22 | 1, 21 | syl 17 | 1 ⊢ (𝑅 ∈ 𝑉 → (ℝ*Hom‘𝑅) = (𝑥 ∈ ℝ* ↦ if(𝑥 ∈ ℝ, ((ℝHom‘𝑅)‘𝑥), if(𝑥 = +∞, (𝑈‘𝐵), (𝐿‘𝐵))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ifcif 4036 ↦ cmpt 4643 “ cima 5041 ‘cfv 5804 ℝcr 9814 +∞cpnf 9950 ℝ*cxr 9952 lubclub 16765 glbcglb 16766 ℝHomcrrh 29365 ℝ*Homcxrh 29388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-xr 9957 df-xrh 29389 |
This theorem is referenced by: (None) |
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