Theorem hsphoival 47219

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

Assertion

Ref	Expression
hsphoival	⊢ (𝜑 → (((𝐻‘𝐴)‘𝐵)‘𝐾) = if(𝐾 ∈ 𝑌, (𝐵‘𝐾), if((𝐵‘𝐾) ≤ 𝐴, (𝐵‘𝐾), 𝐴)))

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

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

Proof of Theorem hsphoival

Step	Hyp	Ref	Expression
1		hsphoival.h	. . . 4 ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝑥, (𝑎‘𝑗), 𝑥)))))
2		breq2 5117	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((𝑎‘𝑗) ≤ 𝑥 ↔ (𝑎‘𝑗) ≤ 𝐴))
3		id 23	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
4	2, 3	ifbieq2d 4519	. . . . . . 7 ⊢ (𝑥 = 𝐴 → if((𝑎‘𝑗) ≤ 𝑥, (𝑎‘𝑗), 𝑥) = if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴))
5	4	ifeq2d 4513	. . . . . 6 ⊢ (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)))
6	5	mpteq2dv 5209	. . . . 5 ⊢ (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))))
7	6	mpteq2dv 5209	. . . 4 ⊢ (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)))))
8		hsphoival.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ)
9		ovex 7444	. . . . . 6 ⊢ (ℝ ↑m 𝑋) ∈ V
10	9	mptex 7222	. . . . 5 ⊢ (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴)))) ∈ V
11	10	a1i 11	. . . 4 ⊢ (𝜑 → (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴)))) ∈ V)
12	1, 7, 8, 11	fvmptd3 7014	. . 3 ⊢ (𝜑 → (𝐻‘𝐴) = (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴)))))
13		fveq1 6881	. . . . . 6 ⊢ (𝑎 = 𝐵 → (𝑎‘𝑗) = (𝐵‘𝑗))
14	13	breq1d 5123	. . . . . . 7 ⊢ (𝑎 = 𝐵 → ((𝑎‘𝑗) ≤ 𝐴 ↔ (𝐵‘𝑗) ≤ 𝐴))
15	14, 13	ifbieq1d 4517	. . . . . 6 ⊢ (𝑎 = 𝐵 → if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴) = if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))
16	13, 15	ifeq12d 4514	. . . . 5 ⊢ (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)))
17	16	mpteq2dv 5209	. . . 4 ⊢ (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))))
18	17	adantl 486	. . 3 ⊢ ((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))))
19		hsphoival.b	. . . 4 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
20		reex 11191	. . . . . . 7 ⊢ ℝ ∈ V
21	20	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ∈ V)
22		hsphoival.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
23	21, 22	jca 520	. . . . 5 ⊢ (𝜑 → (ℝ ∈ V ∧ 𝑋 ∈ 𝑉))
24		elmapg 8836	. . . . 5 ⊢ ((ℝ ∈ V ∧ 𝑋 ∈ 𝑉) → (𝐵 ∈ (ℝ ↑m 𝑋) ↔ 𝐵:𝑋⟶ℝ))
25	23, 24	syl 18	. . . 4 ⊢ (𝜑 → (𝐵 ∈ (ℝ ↑m 𝑋) ↔ 𝐵:𝑋⟶ℝ))
26	19, 25	mpbird 260	. . 3 ⊢ (𝜑 → 𝐵 ∈ (ℝ ↑m 𝑋))
27		mptexg 7220	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))) ∈ V)
28	22, 27	syl 18	. . 3 ⊢ (𝜑 → (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))) ∈ V)
29	12, 18, 26, 28	fvmptd 6998	. 2 ⊢ (𝜑 → ((𝐻‘𝐴)‘𝐵) = (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))))
30		eleq1 2857	. . . 4 ⊢ (𝑗 = 𝐾 → (𝑗 ∈ 𝑌 ↔ 𝐾 ∈ 𝑌))
31		fveq2 6882	. . . 4 ⊢ (𝑗 = 𝐾 → (𝐵‘𝑗) = (𝐵‘𝐾))
32	31	breq1d 5123	. . . . 5 ⊢ (𝑗 = 𝐾 → ((𝐵‘𝑗) ≤ 𝐴 ↔ (𝐵‘𝐾) ≤ 𝐴))
33	32, 31	ifbieq1d 4517	. . . 4 ⊢ (𝑗 = 𝐾 → if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴) = if((𝐵‘𝐾) ≤ 𝐴, (𝐵‘𝐾), 𝐴))
34	30, 31, 33	ifbieq12d 4521	. . 3 ⊢ (𝑗 = 𝐾 → if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴)) = if(𝐾 ∈ 𝑌, (𝐵‘𝐾), if((𝐵‘𝐾) ≤ 𝐴, (𝐵‘𝐾), 𝐴)))
35	34	adantl 486	. 2 ⊢ ((𝜑 ∧ 𝑗 = 𝐾) → if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴)) = if(𝐾 ∈ 𝑌, (𝐵‘𝐾), if((𝐵‘𝐾) ≤ 𝐴, (𝐵‘𝐾), 𝐴)))
36		hsphoival.k	. 2 ⊢ (𝜑 → 𝐾 ∈ 𝑋)
37	19, 36	ffvelcdmd 7081	. . 3 ⊢ (𝜑 → (𝐵‘𝐾) ∈ ℝ)
38	37, 8	ifcld 4539	. . 3 ⊢ (𝜑 → if((𝐵‘𝐾) ≤ 𝐴, (𝐵‘𝐾), 𝐴) ∈ ℝ)
39	37, 38	ifexd 4541	. 2 ⊢ (𝜑 → if(𝐾 ∈ 𝑌, (𝐵‘𝐾), if((𝐵‘𝐾) ≤ 𝐴, (𝐵‘𝐾), 𝐴)) ∈ V)
40	29, 35, 36, 39	fvmptd 6998	1 ⊢ (𝜑 → (((𝐻‘𝐴)‘𝐵)‘𝐾) = if(𝐾 ∈ 𝑌, (𝐵‘𝐾), if((𝐵‘𝐾) ≤ 𝐴, (𝐵‘𝐾), 𝐴)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ifcif 4492   class class class wbr 5113   ↦ cmpt 5196  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411   ↑_m cmap 8824  ℝcr 11099   ≤ cle 11244

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8826

This theorem is referenced by:  hsphoidmvle2  47225  hsphoidmvle  47226  hoidmvlelem2  47236  hspmbllem1  47266