Theorem xrhval 34439

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 3479	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2		fveq2 6888	. . . . . 6 ⊢ (𝑟 = 𝑅 → (ℝHom‘𝑟) = (ℝHom‘𝑅))
3	2	fveq1d 6890	. . . . 5 ⊢ (𝑟 = 𝑅 → ((ℝHom‘𝑟)‘𝑥) = ((ℝHom‘𝑅)‘𝑥))
4		fveq2 6888	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (lub‘𝑟) = (lub‘𝑅))
5		xrhval.u	. . . . . . . 8 ⊢ 𝑈 = (lub‘𝑅)
6	4, 5	eqtr4di 2819	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (lub‘𝑟) = 𝑈)
7	2	imaeq1d 6066	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((ℝHom‘𝑟) “ ℝ) = ((ℝHom‘𝑅) “ ℝ))
8		xrhval.b	. . . . . . . 8 ⊢ 𝐵 = ((ℝHom‘𝑅) “ ℝ)
9	7, 8	eqtr4di 2819	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((ℝHom‘𝑟) “ ℝ) = 𝐵)
10	6, 9	fveq12d 6895	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((lub‘𝑟)‘((ℝHom‘𝑟) “ ℝ)) = (𝑈‘𝐵))
11		fveq2 6888	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (glb‘𝑟) = (glb‘𝑅))
12		xrhval.l	. . . . . . . 8 ⊢ 𝐿 = (glb‘𝑅)
13	11, 12	eqtr4di 2819	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (glb‘𝑟) = 𝐿)
14	13, 9	fveq12d 6895	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((glb‘𝑟)‘((ℝHom‘𝑟) “ ℝ)) = (𝐿‘𝐵))
15	10, 14	ifeq12d 4514	. . . . 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 4514	. . . 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 5210	. . 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 34438	. . 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 13029	. . . 4 ⊢ ℝ* ∈ V
20	19	mptex 7228	. . 3 ⊢ (𝑥 ∈ ℝ* ↦ if(𝑥 ∈ ℝ, ((ℝHom‘𝑅)‘𝑥), if(𝑥 = +∞, (𝑈‘𝐵), (𝐿‘𝐵)))) ∈ V
21	17, 18, 20	fvmpt 6996	. 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 2146  Vcvv 3458  ifcif 4492   ↦ cmpt 5197   “ cima 5669  ‘cfv 6543  ℝcr 11117  +∞cpnf 11258  ℝ^*cxr 11260  lubclub 18390  glbcglb 18391  ℝHomcrrh 34414  ℝ^*Homcxrh 34437

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175

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 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-xr 11265  df-xrh 34438

This theorem is used by: (None)