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Theorem iccpnfhmeo 24928

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 13380	. . . 4 ⊢ (0[,]1) ⊆ ℝ^*
2		xrltso 13089	. . . 4 ⊢ < Or ℝ^*
3		soss 5556	. . . 4 ⊢ ((0[,]1) ⊆ ℝ^* → ( < Or ℝ^* → < Or (0[,]1)))
4	1, 2, 3	mp2 9	. . 3 ⊢ < Or (0[,]1)
5		iccssxr 13380	. . . . 5 ⊢ (0[,]+∞) ⊆ ℝ^*
6		soss 5556	. . . . 5 ⊢ ((0[,]+∞) ⊆ ℝ^* → ( < Or ℝ^* → < Or (0[,]+∞)))
7	5, 2, 6	mp2 9	. . . 4 ⊢ < Or (0[,]+∞)
8		sopo 5555	. . . 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 24927	. . . . 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 6785	. . . 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 13416	. . . . . . . . . . . 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 13416	. . . . . . . . . . . . 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 11142	. . . . . . . . . . 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 11303	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 < 1)
26	17, 25	gtned 11278	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 1 ≠ 𝑧)
27	26	necomd 2988	. . . . . . . 8 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 ≠ 1)
28		ifnefalse 4479	. . . . . . . 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 5090	. . . . . . . 8 ⊢ (+∞ = if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤))) → ((𝑧 / (1 − 𝑧)) < +∞ ↔ (𝑧 / (1 − 𝑧)) < if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤)))))
31		breq2 5090	. . . . . . . 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 11141	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
33		resubcl 11455	. . . . . . . . . . . 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 11093	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
36	17	recnd 11170	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 ∈ ℂ)
37		subeq0 11417	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((1 − 𝑧) = 0 ↔ 1 = 𝑧))
38	37	necon3bid 2977	. . . . . . . . . . . . 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 11980	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (𝑧 / (1 − 𝑧)) ∈ ℝ)
42	41	ltpnfd 13069	. . . . . . . . 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 2737	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))) = (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))
46		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
47	45, 46	icopnfhmeo 24926	. . . . . . . . . . . . 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 11189	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℝ^*
52		1xr 11201	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℝ^*
53		0le1 11670	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ≤ 1
54		snunico 13429	. . . . . . . . . . . . . . . . . . 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 2848	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 ∈ ((0[,)1) ∪ {1}))
57		elun 4094	. . . . . . . . . . . . . . . . 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 4585	. . . . . . . . . . . . . . 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 2950	. . . . . . . . . . . . 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 2848	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑤 ∈ ((0[,)1) ∪ {1}))
67		elun 4094	. . . . . . . . . . . . . . . 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 4585	. . . . . . . . . . . . . 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 7278	. . . . . . . . . . 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 7372	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (1 − 𝑥) = (1 − 𝑧))
79	77, 78	oveq12d 7382	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑥 / (1 − 𝑥)) = (𝑧 / (1 − 𝑧)))
80		ovex 7397	. . . . . . . . . . 11 ⊢ (𝑧 / (1 − 𝑧)) ∈ V
81	79, 45, 80	fvmpt 6945	. . . . . . . . . 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 7372	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (1 − 𝑥) = (1 − 𝑤))
85	83, 84	oveq12d 7382	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → (𝑥 / (1 − 𝑥)) = (𝑤 / (1 − 𝑤)))
86		ovex 7397	. . . . . . . . . . 11 ⊢ (𝑤 / (1 − 𝑤)) ∈ V
87	85, 45, 86	fvmpt 6945	. . . . . . . . . 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 5117	. . . . . . . 8 ⊢ (((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) ∧ ¬ 𝑤 = 1) → (𝑧 / (1 − 𝑧)) < (𝑤 / (1 − 𝑤)))
90	30, 31, 43, 89	ifbothda 4506	. . . . . . 7 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (𝑧 / (1 − 𝑧)) < if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤))))
91	29, 90	eqbrtrd 5108	. . . . . 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 2741	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 = 1 ↔ 𝑧 = 1))
94	93, 79	ifbieq2d 4494	. . . . . . 7 ⊢ (𝑥 = 𝑧 → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) = if(𝑧 = 1, +∞, (𝑧 / (1 − 𝑧))))
95		pnfex 11195	. . . . . . . 8 ⊢ +∞ ∈ V
96	95, 80	ifex 4518	. . . . . . 7 ⊢ if(𝑧 = 1, +∞, (𝑧 / (1 − 𝑧))) ∈ V
97	94, 10, 96	fvmpt 6945	. . . . . 6 ⊢ (𝑧 ∈ (0[,]1) → (𝐹‘𝑧) = if(𝑧 = 1, +∞, (𝑧 / (1 − 𝑧))))
98		eqeq1 2741	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑥 = 1 ↔ 𝑤 = 1))
99	98, 85	ifbieq2d 4494	. . . . . . 7 ⊢ (𝑥 = 𝑤 → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) = if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤))))
100	95, 86	ifex 4518	. . . . . . 7 ⊢ if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤))) ∈ V
101	99, 10, 100	fvmpt 6945	. . . . . 6 ⊢ (𝑤 ∈ (0[,]1) → (𝐹‘𝑤) = if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤))))
102	97, 101	breqan12d 5102	. . . . 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 3178	. . 3 ⊢ ∀𝑧 ∈ (0[,]1)∀𝑤 ∈ (0[,]1)(𝑧 < 𝑤 → (𝐹‘𝑧) < (𝐹‘𝑤))
105		soisoi 7280	. . 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 18556	. . . . . 6 ⊢ ≤ ∈ TosetRel
108	107	elexi 3453	. . . . 5 ⊢ ≤ ∈ V
109	108	inex1 5257	. . . 4 ⊢ ( ≤ ∩ ((0[,]1) × (0[,]1))) ∈ V
110	108	inex1 5257	. . . 4 ⊢ ( ≤ ∩ ((0[,]+∞) × (0[,]+∞))) ∈ V
111		leiso 14418	. . . . . . . 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 7286	. . . . . 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 7285	. . . . 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 18550	. . . . . . . 8 ⊢ ( ≤ ∈ TosetRel → ≤ ∈ PosetRel)
119	107, 118	ax-mp 5	. . . . . . 7 ⊢ ≤ ∈ PosetRel
120		ledm 18553	. . . . . . . 8 ⊢ ℝ^* = dom ≤
121	120	psssdm 18545	. . . . . . 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 2746	. . . . 5 ⊢ (0[,]1) = dom ( ≤ ∩ ((0[,]1) × (0[,]1)))
124	120	psssdm 18545	. . . . . . 7 ⊢ (( ≤ ∈ PosetRel ∧ (0[,]+∞) ⊆ ℝ^*) → dom ( ≤ ∩ ((0[,]+∞) × (0[,]+∞))) = (0[,]+∞))
125	119, 5, 124	mp2an 693	. . . . . 6 ⊢ dom ( ≤ ∩ ((0[,]+∞) × (0[,]+∞))) = (0[,]+∞)
126	125	eqcomi 2746	. . . . 5 ⊢ (0[,]+∞) = dom ( ≤ ∩ ((0[,]+∞) × (0[,]+∞)))
127	123, 126	ordthmeo 23783	. . . 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 24868	. . . 4 ⊢ II = (ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1))))
130		iccpnfhmeo.k	. . . . 5 ⊢ 𝐾 = ((ordTop‘ ≤ ) ↾_t (0[,]+∞))
131		ordtresticc 23204	. . . . 5 ⊢ ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) = (ordTop‘( ≤ ∩ ((0[,]+∞) × (0[,]+∞))))
132	130, 131	eqtri 2760	. . . 4 ⊢ 𝐾 = (ordTop‘( ≤ ∩ ((0[,]+∞) × (0[,]+∞))))
133	129, 132	oveq12i 7376	. . 3 ⊢ (IIHomeo𝐾) = ((ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1))))Homeo(ordTop‘( ≤ ∩ ((0[,]+∞) × (0[,]+∞)))))
134	128, 133	eleqtrri 2836	. 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 2933 ∀wral 3052 Vcvv 3430 ∪ cun 3888 ∩ cin 3889 ⊆ wss 3890 ifcif 4467 {csn 4568 class class class wbr 5086 ↦ cmpt 5167 Po wpo 5534 Or wor 5535 × cxp 5626 ^◡ccnv 5627 dom cdm 5628 –onto→wfo 6494 –1-1-onto→wf1o 6495 ‘cfv 6496 Isom wiso 6497 (class class class)co 7364 ℂcc 11033 ℝcr 11034 0cc0 11035 1c1 11036 + caddc 11038 +∞cpnf 11173 ℝ^*cxr 11175 < clt 11176 ≤ cle 11177 − cmin 11374 / cdiv 11804 [,)cico 13297 [,]cicc 13298 ↾_t crest 17380 TopOpenctopn 17381 ordTopcordt 17460 PosetRelcps 18527 TosetRel ctsr 18528 ℂ_fldccnfld 21350 Homeochmeo 23734 IIcii 24858

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 5213 ax-sep 5232 ax-nul 5242 ax-pow 5306 ax-pr 5374 ax-un 7686 ax-cnex 11091 ax-resscn 11092 ax-1cn 11093 ax-icn 11094 ax-addcl 11095 ax-addrcl 11096 ax-mulcl 11097 ax-mulrcl 11098 ax-mulcom 11099 ax-addass 11100 ax-mulass 11101 ax-distr 11102 ax-i2m1 11103 ax-1ne0 11104 ax-1rid 11105 ax-rnegex 11106 ax-rrecex 11107 ax-cnre 11108 ax-pre-lttri 11109 ax-pre-lttrn 11110 ax-pre-ltadd 11111 ax-pre-mulgt0 11112 ax-pre-sup 11113

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 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5523 df-eprel 5528 df-po 5536 df-so 5537 df-fr 5581 df-we 5583 df-xp 5634 df-rel 5635 df-cnv 5636 df-co 5637 df-dm 5638 df-rn 5639 df-res 5640 df-ima 5641 df-pred 6263 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7321 df-ov 7367 df-oprab 7368 df-mpo 7369 df-om 7815 df-1st 7939 df-2nd 7940 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-er 8640 df-map 8772 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fi 9321 df-sup 9352 df-inf 9353 df-pnf 11178 df-mnf 11179 df-xr 11180 df-ltxr 11181 df-le 11182 df-sub 11376 df-neg 11377 df-div 11805 df-nn 12172 df-2 12241 df-3 12242 df-4 12243 df-5 12244 df-6 12245 df-7 12246 df-8 12247 df-9 12248 df-n0 12435 df-z 12522 df-dec 12642 df-uz 12786 df-q 12896 df-rp 12940 df-xneg 13060 df-xadd 13061 df-xmul 13062 df-ioo 13299 df-ioc 13300 df-ico 13301 df-icc 13302 df-fz 13459 df-seq 13961 df-exp 14021 df-cj 15058 df-re 15059 df-im 15060 df-sqrt 15194 df-abs 15195 df-struct 17114 df-slot 17149 df-ndx 17161 df-base 17177 df-plusg 17230 df-mulr 17231 df-starv 17232 df-tset 17236 df-ple 17237 df-ds 17239 df-unif 17240 df-rest 17382 df-topn 17383 df-topgen 17403 df-ordt 17462 df-ps 18529 df-tsr 18530 df-psmet 21342 df-xmet 21343 df-met 21344 df-bl 21345 df-mopn 21346 df-cnfld 21351 df-top 22875 df-topon 22892 df-topsp 22914 df-bases 22927 df-cn 23208 df-hmeo 23736 df-xms 24301 df-ms 24302 df-ii 24860

This theorem is referenced by: xrhmeo 24929 xrge0hmph 34098

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