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Theorem reparphtiOLD 24965

Description: Obsolete version of reparphti 24964 as of 9-Apr-2025. (Contributed by NM, 15-Jun-2010.) (Revised by Mario Carneiro, 7-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
reparpht.1	⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))
reparpht.2	⊢ (𝜑 → 𝐺 ∈ (II Cn II))
reparpht.3	⊢ (𝜑 → (𝐺‘0) = 0)
reparpht.4	⊢ (𝜑 → (𝐺‘1) = 1)
reparphtiOLD.5	⊢ 𝐻 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))))

**Assertion**
Ref	Expression
reparphtiOLD	⊢ (𝜑 → 𝐻 ∈ ((𝐹 ∘ 𝐺)(PHtpy‘𝐽)𝐹))

Distinct variable groups: 𝑥,𝑦,𝐹 𝑥,𝐺,𝑦 𝑥,𝐽,𝑦 𝜑,𝑥,𝑦 Allowed substitution hints: 𝐻(𝑥,𝑦)

Proof of Theorem reparphtiOLD Dummy variables 𝑠 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reparpht.2	. . 3 ⊢ (𝜑 → 𝐺 ∈ (II Cn II))
2		reparpht.1	. . 3 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))
3		cnco 23222	. . 3 ⊢ ((𝐺 ∈ (II Cn II) ∧ 𝐹 ∈ (II Cn 𝐽)) → (𝐹 ∘ 𝐺) ∈ (II Cn 𝐽))
4	1, 2, 3	syl2anc 585	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ (II Cn 𝐽))
5		reparphtiOLD.5	. . 3 ⊢ 𝐻 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))))
6		iitopon 24840	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
7	6	a1i 11	. . . 4 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1)))
8		eqid 2737	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
9	8	cnfldtop 24739	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) ∈ Top
10		cnrest2r 23243	. . . . . . . . . 10 ⊢ ((TopOpen‘ℂ_fld) ∈ Top → ((II ×_t II) Cn ((TopOpen‘ℂ_fld) ↾_t (0[,]1))) ⊆ ((II ×_t II) Cn (TopOpen‘ℂ_fld)))
11	9, 10	mp1i 13	. . . . . . . . 9 ⊢ (𝜑 → ((II ×_t II) Cn ((TopOpen‘ℂ_fld) ↾_t (0[,]1))) ⊆ ((II ×_t II) Cn (TopOpen‘ℂ_fld)))
12	7, 7	cnmpt2nd 23625	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((II ×_t II) Cn II))
13		iirevcn 24892	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (0[,]1) ↦ (1 − 𝑧)) ∈ (II Cn II)
14	13	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑧 ∈ (0[,]1) ↦ (1 − 𝑧)) ∈ (II Cn II))
15		oveq2 7376	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (1 − 𝑧) = (1 − 𝑦))
16	7, 7, 12, 7, 14, 15	cnmpt21 23627	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (1 − 𝑦)) ∈ ((II ×_t II) Cn II))
17	8	dfii3 24844	. . . . . . . . . . 11 ⊢ II = ((TopOpen‘ℂ_fld) ↾_t (0[,]1))
18	17	oveq2i 7379	. . . . . . . . . 10 ⊢ ((II ×_t II) Cn II) = ((II ×_t II) Cn ((TopOpen‘ℂ_fld) ↾_t (0[,]1)))
19	16, 18	eleqtrdi 2847	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (1 − 𝑦)) ∈ ((II ×_t II) Cn ((TopOpen‘ℂ_fld) ↾_t (0[,]1))))
20	11, 19	sseldd 3936	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (1 − 𝑦)) ∈ ((II ×_t II) Cn (TopOpen‘ℂ_fld)))
21	7, 7	cnmpt1st 23624	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ 𝑥) ∈ ((II ×_t II) Cn II))
22	7, 7, 21, 1	cnmpt21f 23628	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐺‘𝑥)) ∈ ((II ×_t II) Cn II))
23	22, 18	eleqtrdi 2847	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐺‘𝑥)) ∈ ((II ×_t II) Cn ((TopOpen‘ℂ_fld) ↾_t (0[,]1))))
24	11, 23	sseldd 3936	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐺‘𝑥)) ∈ ((II ×_t II) Cn (TopOpen‘ℂ_fld)))
25	8	mulcn 24824	. . . . . . . . 9 ⊢ · ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
26	25	a1i 11	. . . . . . . 8 ⊢ (𝜑 → · ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld)))
27	7, 7, 20, 24, 26	cnmpt22f 23631	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((1 − 𝑦) · (𝐺‘𝑥))) ∈ ((II ×_t II) Cn (TopOpen‘ℂ_fld)))
28	12, 18	eleqtrdi 2847	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((II ×_t II) Cn ((TopOpen‘ℂ_fld) ↾_t (0[,]1))))
29	11, 28	sseldd 3936	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((II ×_t II) Cn (TopOpen‘ℂ_fld)))
30	21, 18	eleqtrdi 2847	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ 𝑥) ∈ ((II ×_t II) Cn ((TopOpen‘ℂ_fld) ↾_t (0[,]1))))
31	11, 30	sseldd 3936	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ 𝑥) ∈ ((II ×_t II) Cn (TopOpen‘ℂ_fld)))
32	7, 7, 29, 31, 26	cnmpt22f 23631	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑦 · 𝑥)) ∈ ((II ×_t II) Cn (TopOpen‘ℂ_fld)))
33	8	addcn 24822	. . . . . . . 8 ⊢ + ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
34	33	a1i 11	. . . . . . 7 ⊢ (𝜑 → + ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld)))
35	7, 7, 27, 32, 34	cnmpt22f 23631	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) ∈ ((II ×_t II) Cn (TopOpen‘ℂ_fld)))
36	8	cnfldtopon 24738	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
37	36	a1i 11	. . . . . . 7 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
38		iiuni 24842	. . . . . . . . . . . . . . 15 ⊢ (0[,]1) = ∪ II
39	38, 38	cnf 23202	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ (II Cn II) → 𝐺:(0[,]1)⟶(0[,]1))
40	1, 39	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺:(0[,]1)⟶(0[,]1))
41	40	ffvelcdmda 7038	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]1)) → (𝐺‘𝑥) ∈ (0[,]1))
42	41	adantrr 718	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1))) → (𝐺‘𝑥) ∈ (0[,]1))
43		simprl 771	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1))) → 𝑥 ∈ (0[,]1))
44		simprr 773	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1))) → 𝑦 ∈ (0[,]1))
45		0re 11146	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
46		1re 11144	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
47		icccvx 24916	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → (((𝐺‘𝑥) ∈ (0[,]1) ∧ 𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) ∈ (0[,]1)))
48	45, 46, 47	mp2an 693	. . . . . . . . . . 11 ⊢ (((𝐺‘𝑥) ∈ (0[,]1) ∧ 𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) ∈ (0[,]1))
49	42, 43, 44, 48	syl3anc 1374	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1))) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) ∈ (0[,]1))
50	49	ralrimivva 3181	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ (0[,]1)∀𝑦 ∈ (0[,]1)(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) ∈ (0[,]1))
51		eqid 2737	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)))
52	51	fmpo 8022	. . . . . . . . 9 ⊢ (∀𝑥 ∈ (0[,]1)∀𝑦 ∈ (0[,]1)(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) ∈ (0[,]1) ↔ (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))):((0[,]1) × (0[,]1))⟶(0[,]1))
53	50, 52	sylib 218	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))):((0[,]1) × (0[,]1))⟶(0[,]1))
54	53	frnd 6678	. . . . . . 7 ⊢ (𝜑 → ran (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) ⊆ (0[,]1))
55		unitssre 13427	. . . . . . . . 9 ⊢ (0[,]1) ⊆ ℝ
56		ax-resscn 11095	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
57	55, 56	sstri 3945	. . . . . . . 8 ⊢ (0[,]1) ⊆ ℂ
58	57	a1i 11	. . . . . . 7 ⊢ (𝜑 → (0[,]1) ⊆ ℂ)
59		cnrest2 23242	. . . . . . 7 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ ran (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) ⊆ (0[,]1) ∧ (0[,]1) ⊆ ℂ) → ((𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) ∈ ((II ×_t II) Cn (TopOpen‘ℂ_fld)) ↔ (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) ∈ ((II ×_t II) Cn ((TopOpen‘ℂ_fld) ↾_t (0[,]1)))))
60	37, 54, 58, 59	syl3anc 1374	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) ∈ ((II ×_t II) Cn (TopOpen‘ℂ_fld)) ↔ (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) ∈ ((II ×_t II) Cn ((TopOpen‘ℂ_fld) ↾_t (0[,]1)))))
61	35, 60	mpbid 232	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) ∈ ((II ×_t II) Cn ((TopOpen‘ℂ_fld) ↾_t (0[,]1))))
62	61, 18	eleqtrrdi 2848	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) ∈ ((II ×_t II) Cn II))
63	7, 7, 62, 2	cnmpt21f 23628	. . 3 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)))) ∈ ((II ×_t II) Cn 𝐽))
64	5, 63	eqeltrid 2841	. 2 ⊢ (𝜑 → 𝐻 ∈ ((II ×_t II) Cn 𝐽))
65	40	ffvelcdmda 7038	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐺‘𝑠) ∈ (0[,]1))
66	57, 65	sselid 3933	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐺‘𝑠) ∈ ℂ)
67	66	mullidd 11162	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1 · (𝐺‘𝑠)) = (𝐺‘𝑠))
68	57	sseli 3931	. . . . . . . 8 ⊢ (𝑠 ∈ (0[,]1) → 𝑠 ∈ ℂ)
69	68	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → 𝑠 ∈ ℂ)
70	69	mul02d 11343	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0 · 𝑠) = 0)
71	67, 70	oveq12d 7386	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 · (𝐺‘𝑠)) + (0 · 𝑠)) = ((𝐺‘𝑠) + 0))
72	66	addridd 11345	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐺‘𝑠) + 0) = (𝐺‘𝑠))
73	71, 72	eqtrd 2772	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 · (𝐺‘𝑠)) + (0 · 𝑠)) = (𝐺‘𝑠))
74	73	fveq2d 6846	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘((1 · (𝐺‘𝑠)) + (0 · 𝑠))) = (𝐹‘(𝐺‘𝑠)))
75		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → 𝑠 ∈ (0[,]1))
76		0elunit 13397	. . . 4 ⊢ 0 ∈ (0[,]1)
77		simpr 484	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑦 = 0)
78	77	oveq2d 7384	. . . . . . . . 9 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (1 − 𝑦) = (1 − 0))
79		1m0e1 12273	. . . . . . . . 9 ⊢ (1 − 0) = 1
80	78, 79	eqtrdi 2788	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (1 − 𝑦) = 1)
81		simpl 482	. . . . . . . . 9 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑥 = 𝑠)
82	81	fveq2d 6846	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (𝐺‘𝑥) = (𝐺‘𝑠))
83	80, 82	oveq12d 7386	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → ((1 − 𝑦) · (𝐺‘𝑥)) = (1 · (𝐺‘𝑠)))
84	77, 81	oveq12d 7386	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (𝑦 · 𝑥) = (0 · 𝑠))
85	83, 84	oveq12d 7386	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) = ((1 · (𝐺‘𝑠)) + (0 · 𝑠)))
86	85	fveq2d 6846	. . . . 5 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) = (𝐹‘((1 · (𝐺‘𝑠)) + (0 · 𝑠))))
87		fvex 6855	. . . . 5 ⊢ (𝐹‘((1 · (𝐺‘𝑠)) + (0 · 𝑠))) ∈ V
88	86, 5, 87	ovmpoa 7523	. . . 4 ⊢ ((𝑠 ∈ (0[,]1) ∧ 0 ∈ (0[,]1)) → (𝑠𝐻0) = (𝐹‘((1 · (𝐺‘𝑠)) + (0 · 𝑠))))
89	75, 76, 88	sylancl 587	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻0) = (𝐹‘((1 · (𝐺‘𝑠)) + (0 · 𝑠))))
90		fvco3 6941	. . . 4 ⊢ ((𝐺:(0[,]1)⟶(0[,]1) ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐺)‘𝑠) = (𝐹‘(𝐺‘𝑠)))
91	40, 90	sylan 581	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐺)‘𝑠) = (𝐹‘(𝐺‘𝑠)))
92	74, 89, 91	3eqtr4d 2782	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻0) = ((𝐹 ∘ 𝐺)‘𝑠))
93		1elunit 13398	. . . 4 ⊢ 1 ∈ (0[,]1)
94		simpr 484	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → 𝑦 = 1)
95	94	oveq2d 7384	. . . . . . . . 9 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (1 − 𝑦) = (1 − 1))
96		1m1e0 12229	. . . . . . . . 9 ⊢ (1 − 1) = 0
97	95, 96	eqtrdi 2788	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (1 − 𝑦) = 0)
98		simpl 482	. . . . . . . . 9 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → 𝑥 = 𝑠)
99	98	fveq2d 6846	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝐺‘𝑥) = (𝐺‘𝑠))
100	97, 99	oveq12d 7386	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ((1 − 𝑦) · (𝐺‘𝑥)) = (0 · (𝐺‘𝑠)))
101	94, 98	oveq12d 7386	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝑦 · 𝑥) = (1 · 𝑠))
102	100, 101	oveq12d 7386	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) = ((0 · (𝐺‘𝑠)) + (1 · 𝑠)))
103	102	fveq2d 6846	. . . . 5 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) = (𝐹‘((0 · (𝐺‘𝑠)) + (1 · 𝑠))))
104		fvex 6855	. . . . 5 ⊢ (𝐹‘((0 · (𝐺‘𝑠)) + (1 · 𝑠))) ∈ V
105	103, 5, 104	ovmpoa 7523	. . . 4 ⊢ ((𝑠 ∈ (0[,]1) ∧ 1 ∈ (0[,]1)) → (𝑠𝐻1) = (𝐹‘((0 · (𝐺‘𝑠)) + (1 · 𝑠))))
106	75, 93, 105	sylancl 587	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻1) = (𝐹‘((0 · (𝐺‘𝑠)) + (1 · 𝑠))))
107	66	mul02d 11343	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0 · (𝐺‘𝑠)) = 0)
108	69	mullidd 11162	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1 · 𝑠) = 𝑠)
109	107, 108	oveq12d 7386	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((0 · (𝐺‘𝑠)) + (1 · 𝑠)) = (0 + 𝑠))
110	69	addlidd 11346	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0 + 𝑠) = 𝑠)
111	109, 110	eqtrd 2772	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((0 · (𝐺‘𝑠)) + (1 · 𝑠)) = 𝑠)
112	111	fveq2d 6846	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘((0 · (𝐺‘𝑠)) + (1 · 𝑠))) = (𝐹‘𝑠))
113	106, 112	eqtrd 2772	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻1) = (𝐹‘𝑠))
114		reparpht.3	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘0) = 0)
115	114	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐺‘0) = 0)
116	115	oveq2d 7384	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · (𝐺‘0)) = ((1 − 𝑠) · 0))
117		ax-1cn 11096	. . . . . . . . 9 ⊢ 1 ∈ ℂ
118		subcl 11391	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ 𝑠 ∈ ℂ) → (1 − 𝑠) ∈ ℂ)
119	117, 69, 118	sylancr 588	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1 − 𝑠) ∈ ℂ)
120	119	mul01d 11344	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · 0) = 0)
121	116, 120	eqtrd 2772	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · (𝐺‘0)) = 0)
122	69	mul01d 11344	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠 · 0) = 0)
123	121, 122	oveq12d 7386	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0)) = (0 + 0))
124		00id 11320	. . . . 5 ⊢ (0 + 0) = 0
125	123, 124	eqtrdi 2788	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0)) = 0)
126	125	fveq2d 6846	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0))) = (𝐹‘0))
127		simpr 484	. . . . . . . . 9 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → 𝑦 = 𝑠)
128	127	oveq2d 7384	. . . . . . . 8 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (1 − 𝑦) = (1 − 𝑠))
129		simpl 482	. . . . . . . . 9 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → 𝑥 = 0)
130	129	fveq2d 6846	. . . . . . . 8 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (𝐺‘𝑥) = (𝐺‘0))
131	128, 130	oveq12d 7386	. . . . . . 7 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → ((1 − 𝑦) · (𝐺‘𝑥)) = ((1 − 𝑠) · (𝐺‘0)))
132	127, 129	oveq12d 7386	. . . . . . 7 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (𝑦 · 𝑥) = (𝑠 · 0))
133	131, 132	oveq12d 7386	. . . . . 6 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) = (((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0)))
134	133	fveq2d 6846	. . . . 5 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) = (𝐹‘(((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0))))
135		fvex 6855	. . . . 5 ⊢ (𝐹‘(((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0))) ∈ V
136	134, 5, 135	ovmpoa 7523	. . . 4 ⊢ ((0 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (𝐹‘(((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0))))
137	76, 75, 136	sylancr 588	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (𝐹‘(((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0))))
138		fvco3 6941	. . . . . 6 ⊢ ((𝐺:(0[,]1)⟶(0[,]1) ∧ 0 ∈ (0[,]1)) → ((𝐹 ∘ 𝐺)‘0) = (𝐹‘(𝐺‘0)))
139	40, 76, 138	sylancl 587	. . . . 5 ⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘0) = (𝐹‘(𝐺‘0)))
140	114	fveq2d 6846	. . . . 5 ⊢ (𝜑 → (𝐹‘(𝐺‘0)) = (𝐹‘0))
141	139, 140	eqtrd 2772	. . . 4 ⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘0) = (𝐹‘0))
142	141	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐺)‘0) = (𝐹‘0))
143	126, 137, 142	3eqtr4d 2782	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = ((𝐹 ∘ 𝐺)‘0))
144		reparpht.4	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘1) = 1)
145	144	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐺‘1) = 1)
146	145	oveq2d 7384	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · (𝐺‘1)) = ((1 − 𝑠) · 1))
147	119	mulridd 11161	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · 1) = (1 − 𝑠))
148	146, 147	eqtrd 2772	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · (𝐺‘1)) = (1 − 𝑠))
149	69	mulridd 11161	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠 · 1) = 𝑠)
150	148, 149	oveq12d 7386	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1)) = ((1 − 𝑠) + 𝑠))
151		npcan 11401	. . . . . 6 ⊢ ((1 ∈ ℂ ∧ 𝑠 ∈ ℂ) → ((1 − 𝑠) + 𝑠) = 1)
152	117, 69, 151	sylancr 588	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) + 𝑠) = 1)
153	150, 152	eqtrd 2772	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1)) = 1)
154	153	fveq2d 6846	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1))) = (𝐹‘1))
155		simpr 484	. . . . . . . . 9 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → 𝑦 = 𝑠)
156	155	oveq2d 7384	. . . . . . . 8 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (1 − 𝑦) = (1 − 𝑠))
157		simpl 482	. . . . . . . . 9 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → 𝑥 = 1)
158	157	fveq2d 6846	. . . . . . . 8 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝐺‘𝑥) = (𝐺‘1))
159	156, 158	oveq12d 7386	. . . . . . 7 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → ((1 − 𝑦) · (𝐺‘𝑥)) = ((1 − 𝑠) · (𝐺‘1)))
160	155, 157	oveq12d 7386	. . . . . . 7 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝑦 · 𝑥) = (𝑠 · 1))
161	159, 160	oveq12d 7386	. . . . . 6 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) = (((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1)))
162	161	fveq2d 6846	. . . . 5 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) = (𝐹‘(((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1))))
163		fvex 6855	. . . . 5 ⊢ (𝐹‘(((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1))) ∈ V
164	162, 5, 163	ovmpoa 7523	. . . 4 ⊢ ((1 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (𝐹‘(((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1))))
165	93, 75, 164	sylancr 588	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (𝐹‘(((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1))))
166		fvco3 6941	. . . . . 6 ⊢ ((𝐺:(0[,]1)⟶(0[,]1) ∧ 1 ∈ (0[,]1)) → ((𝐹 ∘ 𝐺)‘1) = (𝐹‘(𝐺‘1)))
167	40, 93, 166	sylancl 587	. . . . 5 ⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘1) = (𝐹‘(𝐺‘1)))
168	144	fveq2d 6846	. . . . 5 ⊢ (𝜑 → (𝐹‘(𝐺‘1)) = (𝐹‘1))
169	167, 168	eqtrd 2772	. . . 4 ⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘1) = (𝐹‘1))
170	169	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐺)‘1) = (𝐹‘1))
171	154, 165, 170	3eqtr4d 2782	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = ((𝐹 ∘ 𝐺)‘1))
172	4, 2, 64, 92, 113, 143, 171	isphtpy2d 24954	1 ⊢ (𝜑 → 𝐻 ∈ ((𝐹 ∘ 𝐺)(PHtpy‘𝐽)𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ⊆ wss 3903 ↦ cmpt 5181 × cxp 5630 ran crn 5633 ∘ ccom 5636 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 ∈ cmpo 7370 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 − cmin 11376 [,]cicc 13276 ↾_t crest 17352 TopOpenctopn 17353 ℂ_fldccnfld 21321 Topctop 22849 TopOnctopon 22866 Cn ccn 23180 ×_t ctx 23516 IIcii 24836 PHtpycphtpy 24935

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-fi 9326 df-sup 9357 df-inf 9358 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13277 df-icc 13280 df-fz 13436 df-fzo 13583 df-seq 13937 df-exp 13997 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-rest 17354 df-topn 17355 df-0g 17373 df-gsum 17374 df-topgen 17375 df-pt 17376 df-prds 17379 df-xrs 17435 df-qtop 17440 df-imas 17441 df-xps 17443 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-mulg 19010 df-cntz 19258 df-cmn 19723 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-cnfld 21322 df-top 22850 df-topon 22867 df-topsp 22889 df-bases 22902 df-cn 23183 df-cnp 23184 df-tx 23518 df-hmeo 23711 df-xms 24276 df-ms 24277 df-tms 24278 df-ii 24838 df-htpy 24937 df-phtpy 24938

This theorem is referenced by: (None)

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