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 3513 | . 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
2 | fveq2 6665 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (ℝHom‘𝑟) = (ℝHom‘𝑅)) | |
3 | 2 | fveq1d 6667 | . . . . 5 ⊢ (𝑟 = 𝑅 → ((ℝHom‘𝑟)‘𝑥) = ((ℝHom‘𝑅)‘𝑥)) |
4 | fveq2 6665 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (lub‘𝑟) = (lub‘𝑅)) | |
5 | xrhval.u | . . . . . . . 8 ⊢ 𝑈 = (lub‘𝑅) | |
6 | 4, 5 | syl6eqr 2874 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (lub‘𝑟) = 𝑈) |
7 | 2 | imaeq1d 5923 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((ℝHom‘𝑟) “ ℝ) = ((ℝHom‘𝑅) “ ℝ)) |
8 | xrhval.b | . . . . . . . 8 ⊢ 𝐵 = ((ℝHom‘𝑅) “ ℝ) | |
9 | 7, 8 | syl6eqr 2874 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → ((ℝHom‘𝑟) “ ℝ) = 𝐵) |
10 | 6, 9 | fveq12d 6672 | . . . . . 6 ⊢ (𝑟 = 𝑅 → ((lub‘𝑟)‘((ℝHom‘𝑟) “ ℝ)) = (𝑈‘𝐵)) |
11 | fveq2 6665 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (glb‘𝑟) = (glb‘𝑅)) | |
12 | xrhval.l | . . . . . . . 8 ⊢ 𝐿 = (glb‘𝑅) | |
13 | 11, 12 | syl6eqr 2874 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (glb‘𝑟) = 𝐿) |
14 | 13, 9 | fveq12d 6672 | . . . . . 6 ⊢ (𝑟 = 𝑅 → ((glb‘𝑟)‘((ℝHom‘𝑟) “ ℝ)) = (𝐿‘𝐵)) |
15 | 10, 14 | ifeq12d 4487 | . . . . 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 4487 | . . . 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 5155 | . . 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 31253 | . . 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 12380 | . . . 4 ⊢ ℝ* ∈ V | |
20 | 19 | mptex 6980 | . . 3 ⊢ (𝑥 ∈ ℝ* ↦ if(𝑥 ∈ ℝ, ((ℝHom‘𝑅)‘𝑥), if(𝑥 = +∞, (𝑈‘𝐵), (𝐿‘𝐵)))) ∈ V |
21 | 17, 18, 20 | fvmpt 6763 | . 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 1533 ∈ wcel 2110 Vcvv 3495 ifcif 4467 ↦ cmpt 5139 “ cima 5553 ‘cfv 6350 ℝcr 10530 +∞cpnf 10666 ℝ*cxr 10668 lubclub 17546 glbcglb 17547 ℝHomcrrh 31229 ℝ*Homcxrh 31252 |
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-rep 5183 ax-sep 5196 ax-nul 5203 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 |
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-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 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-iun 4914 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-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-xr 10673 df-xrh 31253 |
This theorem is referenced by: (None) |
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