Theorem hsphoif 47026

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 7032	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (𝐵‘𝑗) ∈ ℝ)
3		hsphoif.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
4	3	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → 𝐴 ∈ ℝ)
5	2, 4	ifcld 4508	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴) ∈ ℝ)
6	2, 5	ifcld 4508	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴)) ∈ ℝ)
7		eqid 2740	. . 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 7062	. 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 5083	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((𝑎‘𝑗) ≤ 𝑥 ↔ (𝑎‘𝑗) ≤ 𝐴))
11		id 22	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
12	10, 11	ifbieq2d 4488	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → if((𝑎‘𝑗) ≤ 𝑥, (𝑎‘𝑗), 𝑥) = if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴))
13	12	ifeq2d 4482	. . . . . . 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 5173	. . . . . 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 5173	. . . . 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 7396	. . . . . . 7 ⊢ (ℝ ↑m 𝑋) ∈ V
17	16	mptex 7174	. . . . . 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 6966	. . . 4 ⊢ (𝜑 → (𝐻‘𝐴) = (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴)))))
20		fveq1 6833	. . . . . . 7 ⊢ (𝑎 = 𝐵 → (𝑎‘𝑗) = (𝐵‘𝑗))
21	20	breq1d 5089	. . . . . . . 8 ⊢ (𝑎 = 𝐵 → ((𝑎‘𝑗) ≤ 𝐴 ↔ (𝐵‘𝑗) ≤ 𝐴))
22	21, 20	ifbieq1d 4486	. . . . . . 7 ⊢ (𝑎 = 𝐵 → if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴) = if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))
23	20, 22	ifeq12d 4483	. . . . . 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 5173	. . . . 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 482	. . . 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 11127	. . . . . . . 8 ⊢ ℝ ∈ V
27	26	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ V)
28		hsphoif.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
29	27, 28	jca 516	. . . . . 6 ⊢ (𝜑 → (ℝ ∈ V ∧ 𝑋 ∈ 𝑉))
30		elmapg 8783	. . . . . 6 ⊢ ((ℝ ∈ V ∧ 𝑋 ∈ 𝑉) → (𝐵 ∈ (ℝ ↑m 𝑋) ↔ 𝐵:𝑋⟶ℝ))
31	29, 30	syl 17	. . . . 5 ⊢ (𝜑 → (𝐵 ∈ (ℝ ↑m 𝑋) ↔ 𝐵:𝑋⟶ℝ))
32	1, 31	mpbird 258	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (ℝ ↑m 𝑋))
33		mptexg 7172	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))) ∈ V)
34	28, 33	syl 17	. . . 4 ⊢ (𝜑 → (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))) ∈ V)
35	19, 25, 32, 34	fvmptd 6950	. . 3 ⊢ (𝜑 → ((𝐻‘𝐴)‘𝐵) = (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))))
36	35	feq1d 6644	. 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 258	1 ⊢ (𝜑 → ((𝐻‘𝐴)‘𝐵):𝑋⟶ℝ)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3432  ifcif 4461   class class class wbr 5079   ↦ cmpt 5160  ⟶wf 6488  ‘cfv 6492  (class class class)co 7363   ↑_m cmap 8770  ℝcr 11035   ≤ cle 11178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-map 8772

This theorem is referenced by:  hsphoidmvle2  47035  hsphoidmvle  47036  sge0hsphoire  47039  hoidmvlelem1  47045  hoidmvlelem2  47046  hoidmvlelem4  47048  hspmbllem1  47076  hspmbllem2  47077