Theorem xrhval 33462

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 3492	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2		fveq2 6891	. . . . . 6 ⊢ (𝑟 = 𝑅 → (ℝHom‘𝑟) = (ℝHom‘𝑅))
3	2	fveq1d 6893	. . . . 5 ⊢ (𝑟 = 𝑅 → ((ℝHom‘𝑟)‘𝑥) = ((ℝHom‘𝑅)‘𝑥))
4		fveq2 6891	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (lub‘𝑟) = (lub‘𝑅))
5		xrhval.u	. . . . . . . 8 ⊢ 𝑈 = (lub‘𝑅)
6	4, 5	eqtr4di 2789	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (lub‘𝑟) = 𝑈)
7	2	imaeq1d 6058	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((ℝHom‘𝑟) “ ℝ) = ((ℝHom‘𝑅) “ ℝ))
8		xrhval.b	. . . . . . . 8 ⊢ 𝐵 = ((ℝHom‘𝑅) “ ℝ)
9	7, 8	eqtr4di 2789	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((ℝHom‘𝑟) “ ℝ) = 𝐵)
10	6, 9	fveq12d 6898	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((lub‘𝑟)‘((ℝHom‘𝑟) “ ℝ)) = (𝑈‘𝐵))
11		fveq2 6891	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (glb‘𝑟) = (glb‘𝑅))
12		xrhval.l	. . . . . . . 8 ⊢ 𝐿 = (glb‘𝑅)
13	11, 12	eqtr4di 2789	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (glb‘𝑟) = 𝐿)
14	13, 9	fveq12d 6898	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((glb‘𝑟)‘((ℝHom‘𝑟) “ ℝ)) = (𝐿‘𝐵))
15	10, 14	ifeq12d 4549	. . . . 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 4549	. . . 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 5250	. . 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 33461	. . 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 12978	. . . 4 ⊢ ℝ* ∈ V
20	19	mptex 7227	. . 3 ⊢ (𝑥 ∈ ℝ* ↦ if(𝑥 ∈ ℝ, ((ℝHom‘𝑅)‘𝑥), if(𝑥 = +∞, (𝑈‘𝐵), (𝐿‘𝐵)))) ∈ V
21	17, 18, 20	fvmpt 6998	. 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)))))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105  Vcvv 3473  ifcif 4528   ↦ cmpt 5231   “ cima 5679  ‘cfv 6543  ℝcr 11115  +∞cpnf 11252  ℝ^*cxr 11254  lubclub 18272  glbcglb 18273  ℝHomcrrh 33437  ℝ^*Homcxrh 33460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-xr 11259  df-xrh 33461

This theorem is referenced by: (None)