Theorem hsphoival 40793

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	. . . . 5 ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝑥, (𝑎‘𝑗), 𝑥)))))
2	1	a1i 11	. . . 4 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝑥, (𝑎‘𝑗), 𝑥))))))
3		breq2 4657	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((𝑎‘𝑗) ≤ 𝑥 ↔ (𝑎‘𝑗) ≤ 𝐴))
4		id 22	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
5	3, 4	ifbieq2d 4111	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → if((𝑎‘𝑗) ≤ 𝑥, (𝑎‘𝑗), 𝑥) = if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴))
6	5	ifeq2d 4105	. . . . . . 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)))
7	6	mpteq2dv 4745	. . . . . 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))))
8	7	mpteq2dv 4745	. . . . 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)))))
9	8	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 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)))))
10		hsphoival.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ)
11		ovex 6678	. . . . . 6 ⊢ (ℝ ↑𝑚 𝑋) ∈ V
12	11	mptex 6486	. . . . 5 ⊢ (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴)))) ∈ V
13	12	a1i 11	. . . 4 ⊢ (𝜑 → (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴)))) ∈ V)
14	2, 9, 10, 13	fvmptd 6288	. . 3 ⊢ (𝜑 → (𝐻‘𝐴) = (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴)))))
15		fveq1 6190	. . . . . 6 ⊢ (𝑎 = 𝐵 → (𝑎‘𝑗) = (𝐵‘𝑗))
16	15	breq1d 4663	. . . . . . 7 ⊢ (𝑎 = 𝐵 → ((𝑎‘𝑗) ≤ 𝐴 ↔ (𝐵‘𝑗) ≤ 𝐴))
17	16, 15	ifbieq1d 4109	. . . . . 6 ⊢ (𝑎 = 𝐵 → if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴) = if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))
18	15, 17	ifeq12d 4106	. . . . 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)))
19	18	mpteq2dv 4745	. . . 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))))
20	19	adantl 482	. . 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))))
21		hsphoival.b	. . . 4 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
22		reex 10027	. . . . . . 7 ⊢ ℝ ∈ V
23	22	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ∈ V)
24		hsphoival.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
25	23, 24	jca 554	. . . . 5 ⊢ (𝜑 → (ℝ ∈ V ∧ 𝑋 ∈ 𝑉))
26		elmapg 7870	. . . . 5 ⊢ ((ℝ ∈ V ∧ 𝑋 ∈ 𝑉) → (𝐵 ∈ (ℝ ↑𝑚 𝑋) ↔ 𝐵:𝑋⟶ℝ))
27	25, 26	syl 17	. . . 4 ⊢ (𝜑 → (𝐵 ∈ (ℝ ↑𝑚 𝑋) ↔ 𝐵:𝑋⟶ℝ))
28	21, 27	mpbird 247	. . 3 ⊢ (𝜑 → 𝐵 ∈ (ℝ ↑𝑚 𝑋))
29		mptexg 6484	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))) ∈ V)
30	24, 29	syl 17	. . 3 ⊢ (𝜑 → (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))) ∈ V)
31	14, 20, 28, 30	fvmptd 6288	. 2 ⊢ (𝜑 → ((𝐻‘𝐴)‘𝐵) = (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))))
32		eleq1 2689	. . . 4 ⊢ (𝑗 = 𝐾 → (𝑗 ∈ 𝑌 ↔ 𝐾 ∈ 𝑌))
33		fveq2 6191	. . . 4 ⊢ (𝑗 = 𝐾 → (𝐵‘𝑗) = (𝐵‘𝐾))
34	33	breq1d 4663	. . . . 5 ⊢ (𝑗 = 𝐾 → ((𝐵‘𝑗) ≤ 𝐴 ↔ (𝐵‘𝐾) ≤ 𝐴))
35	34, 33	ifbieq1d 4109	. . . 4 ⊢ (𝑗 = 𝐾 → if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴) = if((𝐵‘𝐾) ≤ 𝐴, (𝐵‘𝐾), 𝐴))
36	32, 33, 35	ifbieq12d 4113	. . 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)))
37	36	adantl 482	. 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)))
38		hsphoival.k	. 2 ⊢ (𝜑 → 𝐾 ∈ 𝑋)
39	21, 38	ffvelrnd 6360	. . 3 ⊢ (𝜑 → (𝐵‘𝐾) ∈ ℝ)
40	39, 10	ifcld 4131	. . 3 ⊢ (𝜑 → if((𝐵‘𝐾) ≤ 𝐴, (𝐵‘𝐾), 𝐴) ∈ ℝ)
41		ifexg 4157	. . 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)
42	39, 40, 41	syl2anc 693	. 2 ⊢ (𝜑 → if(𝐾 ∈ 𝑌, (𝐵‘𝐾), if((𝐵‘𝐾) ≤ 𝐴, (𝐵‘𝐾), 𝐴)) ∈ V)
43	31, 37, 38, 42	fvmptd 6288	1 ⊢ (𝜑 → (((𝐻‘𝐴)‘𝐵)‘𝐾) = if(𝐾 ∈ 𝑌, (𝐵‘𝐾), if((𝐵‘𝐾) ≤ 𝐴, (𝐵‘𝐾), 𝐴)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ifcif 4086   class class class wbr 4653   ↦ cmpt 4729  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857  ℝcr 9935   ≤ cle 10075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859

This theorem is referenced by:  hsphoidmvle2  40799  hsphoidmvle  40800  hoidmvlelem2  40810  hspmbllem1  40840