Theorem hsphoival 42881

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 5070	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((𝑎‘𝑗) ≤ 𝑥 ↔ (𝑎‘𝑗) ≤ 𝐴))
3		id 22	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
4	2, 3	ifbieq2d 4492	. . . . . . 7 ⊢ (𝑥 = 𝐴 → if((𝑎‘𝑗) ≤ 𝑥, (𝑎‘𝑗), 𝑥) = if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴))
5	4	ifeq2d 4486	. . . . . 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 5162	. . . . 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 5162	. . . 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 7189	. . . . . 6 ⊢ (ℝ ↑m 𝑋) ∈ V
10	9	mptex 6986	. . . . 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 6791	. . 3 ⊢ (𝜑 → (𝐻‘𝐴) = (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴)))))
13		fveq1 6669	. . . . . 6 ⊢ (𝑎 = 𝐵 → (𝑎‘𝑗) = (𝐵‘𝑗))
14	13	breq1d 5076	. . . . . . 7 ⊢ (𝑎 = 𝐵 → ((𝑎‘𝑗) ≤ 𝐴 ↔ (𝐵‘𝑗) ≤ 𝐴))
15	14, 13	ifbieq1d 4490	. . . . . 6 ⊢ (𝑎 = 𝐵 → if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴) = if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))
16	13, 15	ifeq12d 4487	. . . . 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 5162	. . . 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 484	. . 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 10628	. . . . . . 7 ⊢ ℝ ∈ V
21	20	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ∈ V)
22		hsphoival.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
23	21, 22	jca 514	. . . . 5 ⊢ (𝜑 → (ℝ ∈ V ∧ 𝑋 ∈ 𝑉))
24		elmapg 8419	. . . . 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 259	. . 3 ⊢ (𝜑 → 𝐵 ∈ (ℝ ↑m 𝑋))
27		mptexg 6984	. . . 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 6775	. 2 ⊢ (𝜑 → ((𝐻‘𝐴)‘𝐵) = (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))))
30		eleq1 2900	. . . 4 ⊢ (𝑗 = 𝐾 → (𝑗 ∈ 𝑌 ↔ 𝐾 ∈ 𝑌))
31		fveq2 6670	. . . 4 ⊢ (𝑗 = 𝐾 → (𝐵‘𝑗) = (𝐵‘𝐾))
32	31	breq1d 5076	. . . . 5 ⊢ (𝑗 = 𝐾 → ((𝐵‘𝑗) ≤ 𝐴 ↔ (𝐵‘𝐾) ≤ 𝐴))
33	32, 31	ifbieq1d 4490	. . . 4 ⊢ (𝑗 = 𝐾 → if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴) = if((𝐵‘𝐾) ≤ 𝐴, (𝐵‘𝐾), 𝐴))
34	30, 31, 33	ifbieq12d 4494	. . 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 484	. 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 6852	. . 3 ⊢ (𝜑 → (𝐵‘𝐾) ∈ ℝ)
38	37, 8	ifcld 4512	. . 3 ⊢ (𝜑 → if((𝐵‘𝐾) ≤ 𝐴, (𝐵‘𝐾), 𝐴) ∈ ℝ)
39		ifexg 4514	. . 3 ⊢ (((𝐵‘𝐾) ∈ ℝ ∧ if((𝐵‘𝐾) ≤ 𝐴, (𝐵‘𝐾), 𝐴) ∈ ℝ) → if(𝐾 ∈ 𝑌, (𝐵‘𝐾), if((𝐵‘𝐾) ≤ 𝐴, (𝐵‘𝐾), 𝐴)) ∈ V)
40	37, 38, 39	syl2anc 586	. 2 ⊢ (𝜑 → if(𝐾 ∈ 𝑌, (𝐵‘𝐾), if((𝐵‘𝐾) ≤ 𝐴, (𝐵‘𝐾), 𝐴)) ∈ V)
41	29, 35, 36, 40	fvmptd 6775	1 ⊢ (𝜑 → (((𝐻‘𝐴)‘𝐵)‘𝐾) = if(𝐾 ∈ 𝑌, (𝐵‘𝐾), if((𝐵‘𝐾) ≤ 𝐴, (𝐵‘𝐾), 𝐴)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3494  ifcif 4467   class class class wbr 5066   ↦ cmpt 5146  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156   ↑_m cmap 8406  ℝcr 10536   ≤ cle 10676

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-map 8408

This theorem is referenced by:  hsphoidmvle2  42887  hsphoidmvle  42888  hoidmvlelem2  42898  hspmbllem1  42928