Theorem xrhval 34733

Description: The value of the embedding from the extended real numbers into a complete lattice. (Contributed by Thierry Arnoux, 19-Feb-2018.)

Hypotheses

Ref	Expression
xrhval.b	⊢ 𝐵 = ((ℝHom‘𝑅) “ ℝ)
xrhval.l	⊢ 𝐿 = (glb‘𝑅)
xrhval.u	⊢ 𝑈 = (lub‘𝑅)

Assertion

Ref	Expression
xrhval	⊢ (𝑅 ∈ 𝑉 → (ℝ*Hom‘𝑅) = (𝑥 ∈ ℝ* ↦ if(𝑥 ∈ ℝ, ((ℝHom‘𝑅)‘𝑥), if(𝑥 = +∞, (𝑈‘𝐵), (𝐿‘𝐵)))))

Distinct variable group:   𝑥,𝑅

Allowed substitution hints:   𝐵(𝑥)   𝑈(𝑥)   𝐿(𝑥)   𝑉(𝑥)

Proof of Theorem xrhval

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3475	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2		fveq2 6911	. . . . . 6 ⊢ (𝑟 = 𝑅 → (ℝHom‘𝑟) = (ℝHom‘𝑅))
3	2	fveq1d 6913	. . . . 5 ⊢ (𝑟 = 𝑅 → ((ℝHom‘𝑟)‘𝑥) = ((ℝHom‘𝑅)‘𝑥))
4		fveq2 6911	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (lub‘𝑟) = (lub‘𝑅))
5		xrhval.u	. . . . . . . 8 ⊢ 𝑈 = (lub‘𝑅)
6	4, 5	eqtr4di 2816	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (lub‘𝑟) = 𝑈)
7	2	imaeq1d 6064	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((ℝHom‘𝑟) “ ℝ) = ((ℝHom‘𝑅) “ ℝ))
8		xrhval.b	. . . . . . . 8 ⊢ 𝐵 = ((ℝHom‘𝑅) “ ℝ)
9	7, 8	eqtr4di 2816	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((ℝHom‘𝑟) “ ℝ) = 𝐵)
10	6, 9	fveq12d 6918	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((lub‘𝑟)‘((ℝHom‘𝑟) “ ℝ)) = (𝑈‘𝐵))
11		fveq2 6911	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (glb‘𝑟) = (glb‘𝑅))
12		xrhval.l	. . . . . . . 8 ⊢ 𝐿 = (glb‘𝑅)
13	11, 12	eqtr4di 2816	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (glb‘𝑟) = 𝐿)
14	13, 9	fveq12d 6918	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((glb‘𝑟)‘((ℝHom‘𝑟) “ ℝ)) = (𝐿‘𝐵))
15	10, 14	ifeq12d 4507	. . . . 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 4507	. . . 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 5203	. . 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 34732	. . 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 13145	. . . 4 ⊢ ℝ* ∈ V
20	19	mptex 7255	. . 3 ⊢ (𝑥 ∈ ℝ* ↦ if(𝑥 ∈ ℝ, ((ℝHom‘𝑅)‘𝑥), if(𝑥 = +∞, (𝑈‘𝐵), (𝐿‘𝐵)))) ∈ V
21	17, 18, 20	fvmpt 7019	. 2 ⊢ (𝑅 ∈ V → (ℝ*Hom‘𝑅) = (𝑥 ∈ ℝ* ↦ if(𝑥 ∈ ℝ, ((ℝHom‘𝑅)‘𝑥), if(𝑥 = +∞, (𝑈‘𝐵), (𝐿‘𝐵)))))
22	1, 21	syl 18	1 ⊢ (𝑅 ∈ 𝑉 → (ℝ*Hom‘𝑅) = (𝑥 ∈ ℝ* ↦ if(𝑥 ∈ ℝ, ((ℝHom‘𝑅)‘𝑥), if(𝑥 = +∞, (𝑈‘𝐵), (𝐿‘𝐵)))))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ifcif 4485   ↦ cmpt 5190   “ cima 5659  ‘cfv 6561  ℝcr 11229  +∞cpnf 11370  ℝ^*cxr 11372  lubclub 18515  glbcglb 18516  ℝHomcrrh 34708  ℝ^*Homcxrh 34731

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pr 5396  ax-un 7778  ax-cnex 11286  ax-resscn 11287

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-xr 11377  df-xrh 34732

This theorem is used by: (None)