Theorem hsphoif 47801

Description: 𝐻 is a function (that returns the representation of the right side of a half-open interval intersected with a half-space). Step (b) in Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)

Hypotheses

Ref	Expression
hsphoif.h	⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝑥, (𝑎‘𝑗), 𝑥)))))
hsphoif.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
hsphoif.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
hsphoif.b	⊢ (𝜑 → 𝐵:𝑋⟶ℝ)

Assertion

Ref	Expression
hsphoif	⊢ (𝜑 → ((𝐻‘𝐴)‘𝐵):𝑋⟶ℝ)

Distinct variable groups:   𝐴,𝑎,𝑗,𝑥   𝐵,𝑎,𝑗   𝑋,𝑎,𝑗,𝑥   𝑌,𝑎,𝑥   𝜑,𝑎,𝑗,𝑥

Allowed substitution hints:   𝐵(𝑥)   𝐻(𝑥, 𝑗, 𝑎)   𝑉(𝑥, 𝑗, 𝑎)   𝑌(𝑗)

Proof of Theorem hsphoif

Step	Hyp	Ref	Expression
1		hsphoif.b	. . . . 5 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
2	1	ffvelcdmda 7111	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (𝐵‘𝑗) ∈ ℝ)
3		hsphoif.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
4	3	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → 𝐴 ∈ ℝ)
5	2, 4	ifcld 4532	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴) ∈ ℝ)
6	2, 5	ifcld 4532	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴)) ∈ ℝ)
7		eqid 2763	. . 3 ⊢ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))) = (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴)))
8	6, 7	fmptd 7141	. 2 ⊢ (𝜑 → (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))):𝑋⟶ℝ)
9		hsphoif.h	. . . . 5 ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝑥, (𝑎‘𝑗), 𝑥)))))
10		breq2 5110	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((𝑎‘𝑗) ≤ 𝑥 ↔ (𝑎‘𝑗) ≤ 𝐴))
11		id 23	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
12	10, 11	ifbieq2d 4512	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → if((𝑎‘𝑗) ≤ 𝑥, (𝑎‘𝑗), 𝑥) = if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴))
13	12	ifeq2d 4506	. . . . . . 7 ⊢ (𝑥 = 𝐴 → if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝑥, (𝑎‘𝑗), 𝑥)) = if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴)))
14	13	mpteq2dv 5203	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝑥, (𝑎‘𝑗), 𝑥))) = (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴))))
15	14	mpteq2dv 5203	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝑥, (𝑎‘𝑗), 𝑥)))) = (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴)))))
16		ovex 7479	. . . . . . 7 ⊢ (ℝ ↑m 𝑋) ∈ V
17	16	mptex 7255	. . . . . 6 ⊢ (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴)))) ∈ V
18	17	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴)))) ∈ V)
19	9, 15, 3, 18	fvmptd3 7043	. . . 4 ⊢ (𝜑 → (𝐻‘𝐴) = (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴)))))
20		fveq1 6910	. . . . . . 7 ⊢ (𝑎 = 𝐵 → (𝑎‘𝑗) = (𝐵‘𝑗))
21	20	breq1d 5116	. . . . . . . 8 ⊢ (𝑎 = 𝐵 → ((𝑎‘𝑗) ≤ 𝐴 ↔ (𝐵‘𝑗) ≤ 𝐴))
22	21, 20	ifbieq1d 4510	. . . . . . 7 ⊢ (𝑎 = 𝐵 → if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴) = if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))
23	20, 22	ifeq12d 4507	. . . . . 6 ⊢ (𝑎 = 𝐵 → if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴)) = if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴)))
24	23	mpteq2dv 5203	. . . . 5 ⊢ (𝑎 = 𝐵 → (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴))) = (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))))
25	24	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝑎 = 𝐵) → (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴))) = (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))))
26		reex 11321	. . . . . . . 8 ⊢ ℝ ∈ V
27	26	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ V)
28		hsphoif.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
29	27, 28	jca 521	. . . . . 6 ⊢ (𝜑 → (ℝ ∈ V ∧ 𝑋 ∈ 𝑉))
30		elmapg 8890	. . . . . 6 ⊢ ((ℝ ∈ V ∧ 𝑋 ∈ 𝑉) → (𝐵 ∈ (ℝ ↑m 𝑋) ↔ 𝐵:𝑋⟶ℝ))
31	29, 30	syl 18	. . . . 5 ⊢ (𝜑 → (𝐵 ∈ (ℝ ↑m 𝑋) ↔ 𝐵:𝑋⟶ℝ))
32	1, 31	mpbird 260	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (ℝ ↑m 𝑋))
33		mptexg 7253	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))) ∈ V)
34	28, 33	syl 18	. . . 4 ⊢ (𝜑 → (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))) ∈ V)
35	19, 25, 32, 34	fvmptd 7027	. . 3 ⊢ (𝜑 → ((𝐻‘𝐴)‘𝐵) = (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))))
36	35	feq1d 6716	. 2 ⊢ (𝜑 → (((𝐻‘𝐴)‘𝐵):𝑋⟶ℝ ↔ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))):𝑋⟶ℝ))
37	8, 36	mpbird 260	1 ⊢ (𝜑 → ((𝐻‘𝐴)‘𝐵):𝑋⟶ℝ)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ifcif 4485   class class class wbr 5106   ↦ cmpt 5190  ⟶wf 6557  ‘cfv 6561  (class class class)co 7446   ↑_m cmap 8878  ℝcr 11229   ≤ cle 11374

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-pow 5332  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-pw 4562  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-ov 7449  df-oprab 7450  df-mpo 7451  df-map 8880

This theorem is used by:  hsphoidmvle2  47810  hsphoidmvle  47811  sge0hsphoire  47814  hoidmvlelem1  47820  hoidmvlelem2  47821  hoidmvlelem4  47823  hspmbllem1  47851  hspmbllem2  47852