Theorem hsphoival 44007

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 5074	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((𝑎‘𝑗) ≤ 𝑥 ↔ (𝑎‘𝑗) ≤ 𝐴))
3		id 22	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
4	2, 3	ifbieq2d 4482	. . . . . . 7 ⊢ (𝑥 = 𝐴 → if((𝑎‘𝑗) ≤ 𝑥, (𝑎‘𝑗), 𝑥) = if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴))
5	4	ifeq2d 4476	. . . . . 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 5172	. . . . 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 5172	. . . 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 7288	. . . . . 6 ⊢ (ℝ ↑m 𝑋) ∈ V
10	9	mptex 7081	. . . . 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 6880	. . 3 ⊢ (𝜑 → (𝐻‘𝐴) = (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴)))))
13		fveq1 6755	. . . . . 6 ⊢ (𝑎 = 𝐵 → (𝑎‘𝑗) = (𝐵‘𝑗))
14	13	breq1d 5080	. . . . . . 7 ⊢ (𝑎 = 𝐵 → ((𝑎‘𝑗) ≤ 𝐴 ↔ (𝐵‘𝑗) ≤ 𝐴))
15	14, 13	ifbieq1d 4480	. . . . . 6 ⊢ (𝑎 = 𝐵 → if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴) = if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))
16	13, 15	ifeq12d 4477	. . . . 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 5172	. . . 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 481	. . 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 10893	. . . . . . 7 ⊢ ℝ ∈ V
21	20	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ∈ V)
22		hsphoival.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
23	21, 22	jca 511	. . . . 5 ⊢ (𝜑 → (ℝ ∈ V ∧ 𝑋 ∈ 𝑉))
24		elmapg 8586	. . . . 5 ⊢ ((ℝ ∈ V ∧ 𝑋 ∈ 𝑉) → (𝐵 ∈ (ℝ ↑m 𝑋) ↔ 𝐵:𝑋⟶ℝ))
25	23, 24	syl 17	. . . 4 ⊢ (𝜑 → (𝐵 ∈ (ℝ ↑m 𝑋) ↔ 𝐵:𝑋⟶ℝ))
26	19, 25	mpbird 256	. . 3 ⊢ (𝜑 → 𝐵 ∈ (ℝ ↑m 𝑋))
27		mptexg 7079	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))) ∈ V)
28	22, 27	syl 17	. . 3 ⊢ (𝜑 → (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))) ∈ V)
29	12, 18, 26, 28	fvmptd 6864	. 2 ⊢ (𝜑 → ((𝐻‘𝐴)‘𝐵) = (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))))
30		eleq1 2826	. . . 4 ⊢ (𝑗 = 𝐾 → (𝑗 ∈ 𝑌 ↔ 𝐾 ∈ 𝑌))
31		fveq2 6756	. . . 4 ⊢ (𝑗 = 𝐾 → (𝐵‘𝑗) = (𝐵‘𝐾))
32	31	breq1d 5080	. . . . 5 ⊢ (𝑗 = 𝐾 → ((𝐵‘𝑗) ≤ 𝐴 ↔ (𝐵‘𝐾) ≤ 𝐴))
33	32, 31	ifbieq1d 4480	. . . 4 ⊢ (𝑗 = 𝐾 → if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴) = if((𝐵‘𝐾) ≤ 𝐴, (𝐵‘𝐾), 𝐴))
34	30, 31, 33	ifbieq12d 4484	. . 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 481	. 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	ffvelrnd 6944	. . 3 ⊢ (𝜑 → (𝐵‘𝐾) ∈ ℝ)
38	37, 8	ifcld 4502	. . 3 ⊢ (𝜑 → if((𝐵‘𝐾) ≤ 𝐴, (𝐵‘𝐾), 𝐴) ∈ ℝ)
39	37, 38	ifexd 4504	. 2 ⊢ (𝜑 → if(𝐾 ∈ 𝑌, (𝐵‘𝐾), if((𝐵‘𝐾) ≤ 𝐴, (𝐵‘𝐾), 𝐴)) ∈ V)
40	29, 35, 36, 39	fvmptd 6864	1 ⊢ (𝜑 → (((𝐻‘𝐴)‘𝐵)‘𝐾) = if(𝐾 ∈ 𝑌, (𝐵‘𝐾), if((𝐵‘𝐾) ≤ 𝐴, (𝐵‘𝐾), 𝐴)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422  ifcif 4456   class class class wbr 5070   ↦ cmpt 5153  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ↑_m cmap 8573  ℝcr 10801   ≤ cle 10941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-map 8575

This theorem is referenced by:  hsphoidmvle2  44013  hsphoidmvle  44014  hoidmvlelem2  44024  hspmbllem1  44054