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Theorem iccpnfhmeo 24901

Description: The defined bijection from [0, 1] to [0, +∞] is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 8-Sep-2015.)

**Hypotheses**
Ref	Expression
iccpnfhmeo.f	⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))))
iccpnfhmeo.k	⊢ 𝐾 = ((ordTop‘ ≤ ) ↾_t (0[,]+∞))

**Assertion**
Ref	Expression
iccpnfhmeo	⊢ (𝐹 Isom < , < ((0[,]1), (0[,]+∞)) ∧ 𝐹 ∈ (IIHomeo𝐾))

Proof of Theorem iccpnfhmeo Dummy variables 𝑦 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iccssxr 13348	. . . 4 ⊢ (0[,]1) ⊆ ℝ^*
2		xrltso 13057	. . . 4 ⊢ < Or ℝ^*
3		soss 5551	. . . 4 ⊢ ((0[,]1) ⊆ ℝ^* → ( < Or ℝ^* → < Or (0[,]1)))
4	1, 2, 3	mp2 9	. . 3 ⊢ < Or (0[,]1)
5		iccssxr 13348	. . . . 5 ⊢ (0[,]+∞) ⊆ ℝ^*
6		soss 5551	. . . . 5 ⊢ ((0[,]+∞) ⊆ ℝ^* → ( < Or ℝ^* → < Or (0[,]+∞)))
7	5, 2, 6	mp2 9	. . . 4 ⊢ < Or (0[,]+∞)
8		sopo 5550	. . . 4 ⊢ ( < Or (0[,]+∞) → < Po (0[,]+∞))
9	7, 8	ax-mp 5	. . 3 ⊢ < Po (0[,]+∞)
10		iccpnfhmeo.f	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))))
11	10	iccpnfcnv 24900	. . . . 5 ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ^◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦)))))
12	11	simpli 483	. . . 4 ⊢ 𝐹:(0[,]1)–1-1-onto→(0[,]+∞)
13		f1ofo 6780	. . . 4 ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) → 𝐹:(0[,]1)–onto→(0[,]+∞))
14	12, 13	ax-mp 5	. . 3 ⊢ 𝐹:(0[,]1)–onto→(0[,]+∞)
15		elicc01 13384	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (0[,]1) ↔ (𝑧 ∈ ℝ ∧ 0 ≤ 𝑧 ∧ 𝑧 ≤ 1))
16	15	simp1bi 1146	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (0[,]1) → 𝑧 ∈ ℝ)
17	16	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 ∈ ℝ)
18		elicc01 13384	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (0[,]1) ↔ (𝑤 ∈ ℝ ∧ 0 ≤ 𝑤 ∧ 𝑤 ≤ 1))
19	18	simp1bi 1146	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (0[,]1) → 𝑤 ∈ ℝ)
20	19	3ad2ant2 1135	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑤 ∈ ℝ)
21		1red 11135	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 1 ∈ ℝ)
22		simp3 1139	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 < 𝑤)
23	18	simp3bi 1148	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (0[,]1) → 𝑤 ≤ 1)
24	23	3ad2ant2 1135	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑤 ≤ 1)
25	17, 20, 21, 22, 24	ltletrd 11295	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 < 1)
26	17, 25	gtned 11270	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 1 ≠ 𝑧)
27	26	necomd 2986	. . . . . . . 8 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 ≠ 1)
28		ifnefalse 4490	. . . . . . . 8 ⊢ (𝑧 ≠ 1 → if(𝑧 = 1, +∞, (𝑧 / (1 − 𝑧))) = (𝑧 / (1 − 𝑧)))
29	27, 28	syl 17	. . . . . . 7 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → if(𝑧 = 1, +∞, (𝑧 / (1 − 𝑧))) = (𝑧 / (1 − 𝑧)))
30		breq2 5101	. . . . . . . 8 ⊢ (+∞ = if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤))) → ((𝑧 / (1 − 𝑧)) < +∞ ↔ (𝑧 / (1 − 𝑧)) < if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤)))))
31		breq2 5101	. . . . . . . 8 ⊢ ((𝑤 / (1 − 𝑤)) = if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤))) → ((𝑧 / (1 − 𝑧)) < (𝑤 / (1 − 𝑤)) ↔ (𝑧 / (1 − 𝑧)) < if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤)))))
32		1re 11134	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
33		resubcl 11447	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (1 − 𝑧) ∈ ℝ)
34	32, 17, 33	sylancr 588	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (1 − 𝑧) ∈ ℝ)
35		ax-1cn 11086	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
36	17	recnd 11162	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 ∈ ℂ)
37		subeq0 11409	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((1 − 𝑧) = 0 ↔ 1 = 𝑧))
38	37	necon3bid 2975	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((1 − 𝑧) ≠ 0 ↔ 1 ≠ 𝑧))
39	35, 36, 38	sylancr 588	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → ((1 − 𝑧) ≠ 0 ↔ 1 ≠ 𝑧))
40	26, 39	mpbird 257	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (1 − 𝑧) ≠ 0)
41	17, 34, 40	redivcld 11971	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (𝑧 / (1 − 𝑧)) ∈ ℝ)
42	41	ltpnfd 13037	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (𝑧 / (1 − 𝑧)) < +∞)
43	42	adantr 480	. . . . . . . 8 ⊢ (((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) ∧ 𝑤 = 1) → (𝑧 / (1 − 𝑧)) < +∞)
44		simpl3 1195	. . . . . . . . . 10 ⊢ (((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) ∧ ¬ 𝑤 = 1) → 𝑧 < 𝑤)
45		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))) = (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))
46		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
47	45, 46	icopnfhmeo 24899	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))) Isom < , < ((0[,)1), (0[,)+∞)) ∧ (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))) ∈ (((TopOpen‘ℂ_fld) ↾_t (0[,)1))Homeo((TopOpen‘ℂ_fld) ↾_t (0[,)+∞))))
48	47	simpli 483	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))) Isom < , < ((0[,)1), (0[,)+∞))
49	48	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) ∧ ¬ 𝑤 = 1) → (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))) Isom < , < ((0[,)1), (0[,)+∞)))
50		simp1 1137	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 ∈ (0[,]1))
51		0xr 11181	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℝ^*
52		1xr 11193	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℝ^*
53		0le1 11662	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ≤ 1
54		snunico 13397	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ 0 ≤ 1) → ((0[,)1) ∪ {1}) = (0[,]1))
55	51, 52, 53, 54	mp3an 1464	. . . . . . . . . . . . . . . . . 18 ⊢ ((0[,)1) ∪ {1}) = (0[,]1)
56	50, 55	eleqtrrdi 2846	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 ∈ ((0[,)1) ∪ {1}))
57		elun 4104	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ((0[,)1) ∪ {1}) ↔ (𝑧 ∈ (0[,)1) ∨ 𝑧 ∈ {1}))
58	56, 57	sylib 218	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (𝑧 ∈ (0[,)1) ∨ 𝑧 ∈ {1}))
59	58	ord 865	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (¬ 𝑧 ∈ (0[,)1) → 𝑧 ∈ {1}))
60		elsni 4596	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ {1} → 𝑧 = 1)
61	59, 60	syl6 35	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (¬ 𝑧 ∈ (0[,)1) → 𝑧 = 1))
62	61	necon1ad 2948	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (𝑧 ≠ 1 → 𝑧 ∈ (0[,)1)))
63	27, 62	mpd 15	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 ∈ (0[,)1))
64	63	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) ∧ ¬ 𝑤 = 1) → 𝑧 ∈ (0[,)1))
65		simp2 1138	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑤 ∈ (0[,]1))
66	65, 55	eleqtrrdi 2846	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑤 ∈ ((0[,)1) ∪ {1}))
67		elun 4104	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ((0[,)1) ∪ {1}) ↔ (𝑤 ∈ (0[,)1) ∨ 𝑤 ∈ {1}))
68	66, 67	sylib 218	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (𝑤 ∈ (0[,)1) ∨ 𝑤 ∈ {1}))
69	68	ord 865	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (¬ 𝑤 ∈ (0[,)1) → 𝑤 ∈ {1}))
70		elsni 4596	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ {1} → 𝑤 = 1)
71	69, 70	syl6 35	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (¬ 𝑤 ∈ (0[,)1) → 𝑤 = 1))
72	71	con1d 145	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (¬ 𝑤 = 1 → 𝑤 ∈ (0[,)1)))
73	72	imp 406	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) ∧ ¬ 𝑤 = 1) → 𝑤 ∈ (0[,)1))
74		isorel 7272	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))) Isom < , < ((0[,)1), (0[,)+∞)) ∧ (𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1))) → (𝑧 < 𝑤 ↔ ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑧) < ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑤)))
75	49, 64, 73, 74	syl12anc 837	. . . . . . . . . 10 ⊢ (((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) ∧ ¬ 𝑤 = 1) → (𝑧 < 𝑤 ↔ ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑧) < ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑤)))
76	44, 75	mpbid 232	. . . . . . . . 9 ⊢ (((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) ∧ ¬ 𝑤 = 1) → ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑧) < ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑤))
77		id 22	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
78		oveq2 7366	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (1 − 𝑥) = (1 − 𝑧))
79	77, 78	oveq12d 7376	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑥 / (1 − 𝑥)) = (𝑧 / (1 − 𝑧)))
80		ovex 7391	. . . . . . . . . . 11 ⊢ (𝑧 / (1 − 𝑧)) ∈ V
81	79, 45, 80	fvmpt 6940	. . . . . . . . . 10 ⊢ (𝑧 ∈ (0[,)1) → ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑧) = (𝑧 / (1 − 𝑧)))
82	64, 81	syl 17	. . . . . . . . 9 ⊢ (((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) ∧ ¬ 𝑤 = 1) → ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑧) = (𝑧 / (1 − 𝑧)))
83		id 22	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → 𝑥 = 𝑤)
84		oveq2 7366	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (1 − 𝑥) = (1 − 𝑤))
85	83, 84	oveq12d 7376	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → (𝑥 / (1 − 𝑥)) = (𝑤 / (1 − 𝑤)))
86		ovex 7391	. . . . . . . . . . 11 ⊢ (𝑤 / (1 − 𝑤)) ∈ V
87	85, 45, 86	fvmpt 6940	. . . . . . . . . 10 ⊢ (𝑤 ∈ (0[,)1) → ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑤) = (𝑤 / (1 − 𝑤)))
88	73, 87	syl 17	. . . . . . . . 9 ⊢ (((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) ∧ ¬ 𝑤 = 1) → ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑤) = (𝑤 / (1 − 𝑤)))
89	76, 82, 88	3brtr3d 5128	. . . . . . . 8 ⊢ (((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) ∧ ¬ 𝑤 = 1) → (𝑧 / (1 − 𝑧)) < (𝑤 / (1 − 𝑤)))
90	30, 31, 43, 89	ifbothda 4517	. . . . . . 7 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (𝑧 / (1 − 𝑧)) < if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤))))
91	29, 90	eqbrtrd 5119	. . . . . 6 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → if(𝑧 = 1, +∞, (𝑧 / (1 − 𝑧))) < if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤))))
92	91	3expia 1122	. . . . 5 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1)) → (𝑧 < 𝑤 → if(𝑧 = 1, +∞, (𝑧 / (1 − 𝑧))) < if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤)))))
93		eqeq1 2739	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 = 1 ↔ 𝑧 = 1))
94	93, 79	ifbieq2d 4505	. . . . . . 7 ⊢ (𝑥 = 𝑧 → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) = if(𝑧 = 1, +∞, (𝑧 / (1 − 𝑧))))
95		pnfex 11187	. . . . . . . 8 ⊢ +∞ ∈ V
96	95, 80	ifex 4529	. . . . . . 7 ⊢ if(𝑧 = 1, +∞, (𝑧 / (1 − 𝑧))) ∈ V
97	94, 10, 96	fvmpt 6940	. . . . . 6 ⊢ (𝑧 ∈ (0[,]1) → (𝐹‘𝑧) = if(𝑧 = 1, +∞, (𝑧 / (1 − 𝑧))))
98		eqeq1 2739	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑥 = 1 ↔ 𝑤 = 1))
99	98, 85	ifbieq2d 4505	. . . . . . 7 ⊢ (𝑥 = 𝑤 → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) = if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤))))
100	95, 86	ifex 4529	. . . . . . 7 ⊢ if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤))) ∈ V
101	99, 10, 100	fvmpt 6940	. . . . . 6 ⊢ (𝑤 ∈ (0[,]1) → (𝐹‘𝑤) = if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤))))
102	97, 101	breqan12d 5113	. . . . 5 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1)) → ((𝐹‘𝑧) < (𝐹‘𝑤) ↔ if(𝑧 = 1, +∞, (𝑧 / (1 − 𝑧))) < if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤)))))
103	92, 102	sylibrd 259	. . . 4 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1)) → (𝑧 < 𝑤 → (𝐹‘𝑧) < (𝐹‘𝑤)))
104	103	rgen2 3175	. . 3 ⊢ ∀𝑧 ∈ (0[,]1)∀𝑤 ∈ (0[,]1)(𝑧 < 𝑤 → (𝐹‘𝑧) < (𝐹‘𝑤))
105		soisoi 7274	. . 3 ⊢ ((( < Or (0[,]1) ∧ < Po (0[,]+∞)) ∧ (𝐹:(0[,]1)–onto→(0[,]+∞) ∧ ∀𝑧 ∈ (0[,]1)∀𝑤 ∈ (0[,]1)(𝑧 < 𝑤 → (𝐹‘𝑧) < (𝐹‘𝑤)))) → 𝐹 Isom < , < ((0[,]1), (0[,]+∞)))
106	4, 9, 14, 104, 105	mp4an 694	. 2 ⊢ 𝐹 Isom < , < ((0[,]1), (0[,]+∞))
107		letsr 18518	. . . . . 6 ⊢ ≤ ∈ TosetRel
108	107	elexi 3462	. . . . 5 ⊢ ≤ ∈ V
109	108	inex1 5261	. . . 4 ⊢ ( ≤ ∩ ((0[,]1) × (0[,]1))) ∈ V
110	108	inex1 5261	. . . 4 ⊢ ( ≤ ∩ ((0[,]+∞) × (0[,]+∞))) ∈ V
111		leiso 14384	. . . . . . . 8 ⊢ (((0[,]1) ⊆ ℝ^* ∧ (0[,]+∞) ⊆ ℝ^*) → (𝐹 Isom < , < ((0[,]1), (0[,]+∞)) ↔ 𝐹 Isom ≤ , ≤ ((0[,]1), (0[,]+∞))))
112	1, 5, 111	mp2an 693	. . . . . . 7 ⊢ (𝐹 Isom < , < ((0[,]1), (0[,]+∞)) ↔ 𝐹 Isom ≤ , ≤ ((0[,]1), (0[,]+∞)))
113	106, 112	mpbi 230	. . . . . 6 ⊢ 𝐹 Isom ≤ , ≤ ((0[,]1), (0[,]+∞))
114		isores1 7280	. . . . . 6 ⊢ (𝐹 Isom ≤ , ≤ ((0[,]1), (0[,]+∞)) ↔ 𝐹 Isom ( ≤ ∩ ((0[,]1) × (0[,]1))), ≤ ((0[,]1), (0[,]+∞)))
115	113, 114	mpbi 230	. . . . 5 ⊢ 𝐹 Isom ( ≤ ∩ ((0[,]1) × (0[,]1))), ≤ ((0[,]1), (0[,]+∞))
116		isores2 7279	. . . . 5 ⊢ (𝐹 Isom ( ≤ ∩ ((0[,]1) × (0[,]1))), ≤ ((0[,]1), (0[,]+∞)) ↔ 𝐹 Isom ( ≤ ∩ ((0[,]1) × (0[,]1))), ( ≤ ∩ ((0[,]+∞) × (0[,]+∞)))((0[,]1), (0[,]+∞)))
117	115, 116	mpbi 230	. . . 4 ⊢ 𝐹 Isom ( ≤ ∩ ((0[,]1) × (0[,]1))), ( ≤ ∩ ((0[,]+∞) × (0[,]+∞)))((0[,]1), (0[,]+∞))
118		tsrps 18512	. . . . . . . 8 ⊢ ( ≤ ∈ TosetRel → ≤ ∈ PosetRel)
119	107, 118	ax-mp 5	. . . . . . 7 ⊢ ≤ ∈ PosetRel
120		ledm 18515	. . . . . . . 8 ⊢ ℝ^* = dom ≤
121	120	psssdm 18507	. . . . . . 7 ⊢ (( ≤ ∈ PosetRel ∧ (0[,]1) ⊆ ℝ^*) → dom ( ≤ ∩ ((0[,]1) × (0[,]1))) = (0[,]1))
122	119, 1, 121	mp2an 693	. . . . . 6 ⊢ dom ( ≤ ∩ ((0[,]1) × (0[,]1))) = (0[,]1)
123	122	eqcomi 2744	. . . . 5 ⊢ (0[,]1) = dom ( ≤ ∩ ((0[,]1) × (0[,]1)))
124	120	psssdm 18507	. . . . . . 7 ⊢ (( ≤ ∈ PosetRel ∧ (0[,]+∞) ⊆ ℝ^*) → dom ( ≤ ∩ ((0[,]+∞) × (0[,]+∞))) = (0[,]+∞))
125	119, 5, 124	mp2an 693	. . . . . 6 ⊢ dom ( ≤ ∩ ((0[,]+∞) × (0[,]+∞))) = (0[,]+∞)
126	125	eqcomi 2744	. . . . 5 ⊢ (0[,]+∞) = dom ( ≤ ∩ ((0[,]+∞) × (0[,]+∞)))
127	123, 126	ordthmeo 23748	. . . 4 ⊢ ((( ≤ ∩ ((0[,]1) × (0[,]1))) ∈ V ∧ ( ≤ ∩ ((0[,]+∞) × (0[,]+∞))) ∈ V ∧ 𝐹 Isom ( ≤ ∩ ((0[,]1) × (0[,]1))), ( ≤ ∩ ((0[,]+∞) × (0[,]+∞)))((0[,]1), (0[,]+∞))) → 𝐹 ∈ ((ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1))))Homeo(ordTop‘( ≤ ∩ ((0[,]+∞) × (0[,]+∞))))))
128	109, 110, 117, 127	mp3an 1464	. . 3 ⊢ 𝐹 ∈ ((ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1))))Homeo(ordTop‘( ≤ ∩ ((0[,]+∞) × (0[,]+∞)))))
129		dfii5 24836	. . . 4 ⊢ II = (ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1))))
130		iccpnfhmeo.k	. . . . 5 ⊢ 𝐾 = ((ordTop‘ ≤ ) ↾_t (0[,]+∞))
131		ordtresticc 23169	. . . . 5 ⊢ ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) = (ordTop‘( ≤ ∩ ((0[,]+∞) × (0[,]+∞))))
132	130, 131	eqtri 2758	. . . 4 ⊢ 𝐾 = (ordTop‘( ≤ ∩ ((0[,]+∞) × (0[,]+∞))))
133	129, 132	oveq12i 7370	. . 3 ⊢ (IIHomeo𝐾) = ((ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1))))Homeo(ordTop‘( ≤ ∩ ((0[,]+∞) × (0[,]+∞)))))
134	128, 133	eleqtrri 2834	. 2 ⊢ 𝐹 ∈ (IIHomeo𝐾)
135	106, 134	pm3.2i 470	1 ⊢ (𝐹 Isom < , < ((0[,]1), (0[,]+∞)) ∧ 𝐹 ∈ (IIHomeo𝐾))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2931 ∀wral 3050 Vcvv 3439 ∪ cun 3898 ∩ cin 3899 ⊆ wss 3900 ifcif 4478 {csn 4579 class class class wbr 5097 ↦ cmpt 5178 Po wpo 5529 Or wor 5530 × cxp 5621 ^◡ccnv 5622 dom cdm 5623 –onto→wfo 6489 –1-1-onto→wf1o 6490 ‘cfv 6491 Isom wiso 6492 (class class class)co 7358 ℂcc 11026 ℝcr 11027 0cc0 11028 1c1 11029 + caddc 11031 +∞cpnf 11165 ℝ^*cxr 11167 < clt 11168 ≤ cle 11169 − cmin 11366 / cdiv 11796 [,)cico 13265 [,]cicc 13266 ↾_t crest 17342 TopOpenctopn 17343 ordTopcordt 17422 PosetRelcps 18489 TosetRel ctsr 18490 ℂ_fldccnfld 21311 Homeochmeo 23699 IIcii 24826

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 2183 ax-ext 2707 ax-rep 5223 ax-sep 5240 ax-nul 5250 ax-pow 5309 ax-pr 5376 ax-un 7680 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

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 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3349 df-reu 3350 df-rab 3399 df-v 3441 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4285 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4902 df-iun 4947 df-iin 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6258 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6447 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-isom 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8767 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-fi 9316 df-sup 9347 df-inf 9348 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-z 12491 df-dec 12610 df-uz 12754 df-q 12864 df-rp 12908 df-xneg 13028 df-xadd 13029 df-xmul 13030 df-ioo 13267 df-ioc 13268 df-ico 13269 df-icc 13270 df-fz 13426 df-seq 13927 df-exp 13987 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-struct 17076 df-slot 17111 df-ndx 17123 df-base 17139 df-plusg 17192 df-mulr 17193 df-starv 17194 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-rest 17344 df-topn 17345 df-topgen 17365 df-ordt 17424 df-ps 18491 df-tsr 18492 df-psmet 21303 df-xmet 21304 df-met 21305 df-bl 21306 df-mopn 21307 df-cnfld 21312 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22892 df-cn 23173 df-hmeo 23701 df-xms 24266 df-ms 24267 df-ii 24828

This theorem is referenced by: xrhmeo 24902 xrge0hmph 34068

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