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Theorem iccpnfhmeo 25293

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 13585	. . . 4 ⊢ (0[,]1) ⊆ ℝ^*
2		xrltso 13294	. . . 4 ⊢ < Or ℝ^*
3		soss 5583	. . . 4 ⊢ ((0[,]1) ⊆ ℝ^* → ( < Or ℝ^* → < Or (0[,]1)))
4	1, 2, 3	mp2 9	. . 3 ⊢ < Or (0[,]1)
5		iccssxr 13585	. . . . 5 ⊢ (0[,]+∞) ⊆ ℝ^*
6		soss 5583	. . . . 5 ⊢ ((0[,]+∞) ⊆ ℝ^* → ( < Or ℝ^* → < Or (0[,]+∞)))
7	5, 2, 6	mp2 9	. . . 4 ⊢ < Or (0[,]+∞)
8		sopo 5582	. . . 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 25292	. . . . 5 ⊢ (𝐹:(0[,]1)⟶1-1onto(0[,]+∞) ∧ ^◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦)))))
12	11	simpli 489	. . . 4 ⊢ 𝐹:(0[,]1)⟶1-1onto(0[,]+∞)
13		f1ofo 6851	. . . 4 ⊢ (𝐹:(0[,]1)⟶1-1onto(0[,]+∞) → 𝐹:(0[,]1)⟶onto(0[,]+∞))
14	12, 13	ax-mp 5	. . 3 ⊢ 𝐹:(0[,]1)⟶onto(0[,]+∞)
15		elicc01 13621	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (0[,]1) ↔ (𝑧 ∈ ℝ ∧ 0 ≤ 𝑧 ∧ 𝑧 ≤ 1))
16	15	simp1bi 1163	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (0[,]1) → 𝑧 ∈ ℝ)
17	16	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 ∈ ℝ)
18		elicc01 13621	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (0[,]1) ↔ (𝑤 ∈ ℝ ∧ 0 ≤ 𝑤 ∧ 𝑤 ≤ 1))
19	18	simp1bi 1163	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (0[,]1) → 𝑤 ∈ ℝ)
20	19	3ad2ant2 1152	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑤 ∈ ℝ)
21		1red 11333	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 1 ∈ ℝ)
22		simp3 1156	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 < 𝑤)
23	18	simp3bi 1165	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (0[,]1) → 𝑤 ≤ 1)
24	23	3ad2ant2 1152	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑤 ≤ 1)
25	17, 20, 21, 22, 24	ltletrd 11494	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 < 1)
26	17, 25	gtned 11469	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 1 ≠ 𝑧)
27	26	necomd 3014	. . . . . . . 8 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 ≠ 1)
28		ifnefalse 4497	. . . . . . . 8 ⊢ (𝑧 ≠ 1 → if(𝑧 = 1, +∞, (𝑧 / (1 − 𝑧))) = (𝑧 / (1 − 𝑧)))
29	27, 28	syl 18	. . . . . . 7 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → if(𝑧 = 1, +∞, (𝑧 / (1 − 𝑧))) = (𝑧 / (1 − 𝑧)))
30		breq2 5110	. . . . . . . 8 ⊢ (+∞ = if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤))) → ((𝑧 / (1 − 𝑧)) < +∞ ↔ (𝑧 / (1 − 𝑧)) < if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤)))))
31		breq2 5110	. . . . . . . 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 11332	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
33		resubcl 11646	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (1 − 𝑧) ∈ ℝ)
34	32, 17, 33	sylancr 599	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (1 − 𝑧) ∈ ℝ)
35		ax-1cn 11282	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
36	17	recnd 11361	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 ∈ ℂ)
37		subeq0 11608	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((1 − 𝑧) = 0 ↔ 1 = 𝑧))
38	37	necon3bid 3003	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((1 − 𝑧) ≠ 0 ↔ 1 ≠ 𝑧))
39	35, 36, 38	sylancr 599	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → ((1 − 𝑧) ≠ 0 ↔ 1 ≠ 𝑧))
40	26, 39	mpbird 260	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (1 − 𝑧) ≠ 0)
41	17, 34, 40	redivcld 12169	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (𝑧 / (1 − 𝑧)) ∈ ℝ)
42	41	ltpnfd 13274	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (𝑧 / (1 − 𝑧)) < +∞)
43	42	adantr 486	. . . . . . . 8 ⊢ (((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) ∧ 𝑤 = 1) → (𝑧 / (1 − 𝑧)) < +∞)
44		simpl3 1212	. . . . . . . . . 10 ⊢ (((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) ∧ ¬ 𝑤 = 1) → 𝑧 < 𝑤)
45		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))) = (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))
46		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
47	45, 46	icopnfhmeo 25291	. . . . . . . . . . . . 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 489	. . . . . . . . . . . 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 1154	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 ∈ (0[,]1))
51		0xr 11380	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℝ^*
52		1xr 11392	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℝ^*
53		0le1 11863	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ≤ 1
54		snunico 13634	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ 0 ≤ 1) → ((0[,)1) ∪ {1}) = (0[,]1))
55	51, 52, 53, 54	mp3an 1491	. . . . . . . . . . . . . . . . . 18 ⊢ ((0[,)1) ∪ {1}) = (0[,]1)
56	50, 55	eleqtrrdi 2875	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 ∈ ((0[,)1) ∪ {1}))
57		elun 4155	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ((0[,)1) ∪ {1}) ↔ (𝑧 ∈ (0[,)1) ∨ 𝑧 ∈ {1}))
58	56, 57	sylib 221	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (𝑧 ∈ (0[,)1) ∨ 𝑧 ∈ {1}))
59	58	ord 878	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (¬ 𝑧 ∈ (0[,)1) → 𝑧 ∈ {1}))
60		elsni 4601	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ {1} → 𝑧 = 1)
61	59, 60	syl6 36	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (¬ 𝑧 ∈ (0[,)1) → 𝑧 = 1))
62	61	necon1ad 2976	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (𝑧 ≠ 1 → 𝑧 ∈ (0[,)1)))
63	27, 62	mpd 16	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 ∈ (0[,)1))
64	63	adantr 486	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) ∧ ¬ 𝑤 = 1) → 𝑧 ∈ (0[,)1))
65		simp2 1155	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑤 ∈ (0[,]1))
66	65, 55	eleqtrrdi 2875	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑤 ∈ ((0[,)1) ∪ {1}))
67		elun 4155	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ((0[,)1) ∪ {1}) ↔ (𝑤 ∈ (0[,)1) ∨ 𝑤 ∈ {1}))
68	66, 67	sylib 221	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (𝑤 ∈ (0[,)1) ∨ 𝑤 ∈ {1}))
69	68	ord 878	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (¬ 𝑤 ∈ (0[,)1) → 𝑤 ∈ {1}))
70		elsni 4601	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ {1} → 𝑤 = 1)
71	69, 70	syl6 36	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (¬ 𝑤 ∈ (0[,)1) → 𝑤 = 1))
72	71	con1d 146	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (¬ 𝑤 = 1 → 𝑤 ∈ (0[,)1)))
73	72	imp 412	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) ∧ ¬ 𝑤 = 1) → 𝑤 ∈ (0[,)1))
74		isorel 7354	. . . . . . . . . . 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 850	. . . . . . . . . 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 235	. . . . . . . . 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 23	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
78		oveq2 7448	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (1 − 𝑥) = (1 − 𝑧))
79	77, 78	oveq12d 7458	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑥 / (1 − 𝑥)) = (𝑧 / (1 − 𝑧)))
80		ovex 7473	. . . . . . . . . . 11 ⊢ (𝑧 / (1 − 𝑧)) ∈ V
81	79, 45, 80	fvmpt 7013	. . . . . . . . . 10 ⊢ (𝑧 ∈ (0[,)1) → ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑧) = (𝑧 / (1 − 𝑧)))
82	64, 81	syl 18	. . . . . . . . 9 ⊢ (((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) ∧ ¬ 𝑤 = 1) → ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑧) = (𝑧 / (1 − 𝑧)))
83		id 23	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → 𝑥 = 𝑤)
84		oveq2 7448	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (1 − 𝑥) = (1 − 𝑤))
85	83, 84	oveq12d 7458	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → (𝑥 / (1 − 𝑥)) = (𝑤 / (1 − 𝑤)))
86		ovex 7473	. . . . . . . . . . 11 ⊢ (𝑤 / (1 − 𝑤)) ∈ V
87	85, 45, 86	fvmpt 7013	. . . . . . . . . 10 ⊢ (𝑤 ∈ (0[,)1) → ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑤) = (𝑤 / (1 − 𝑤)))
88	73, 87	syl 18	. . . . . . . . 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 5139	. . . . . . . 8 ⊢ (((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) ∧ ¬ 𝑤 = 1) → (𝑧 / (1 − 𝑧)) < (𝑤 / (1 − 𝑤)))
90	30, 31, 43, 89	ifbothda 4524	. . . . . . 7 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (𝑧 / (1 − 𝑧)) < if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤))))
91	29, 90	eqbrtrd 5130	. . . . . 6 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → if(𝑧 = 1, +∞, (𝑧 / (1 − 𝑧))) < if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤))))
92	91	3expia 1139	. . . . 5 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1)) → (𝑧 < 𝑤 → if(𝑧 = 1, +∞, (𝑧 / (1 − 𝑧))) < if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤)))))
93		eqeq1 2767	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 = 1 ↔ 𝑧 = 1))
94	93, 79	ifbieq2d 4512	. . . . . . 7 ⊢ (𝑥 = 𝑧 → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) = if(𝑧 = 1, +∞, (𝑧 / (1 − 𝑧))))
95		pnfex 11386	. . . . . . . 8 ⊢ +∞ ∈ V
96	95, 80	ifex 4536	. . . . . . 7 ⊢ if(𝑧 = 1, +∞, (𝑧 / (1 − 𝑧))) ∈ V
97	94, 10, 96	fvmpt 7013	. . . . . 6 ⊢ (𝑧 ∈ (0[,]1) → (𝐹‘𝑧) = if(𝑧 = 1, +∞, (𝑧 / (1 − 𝑧))))
98		eqeq1 2767	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑥 = 1 ↔ 𝑤 = 1))
99	98, 85	ifbieq2d 4512	. . . . . . 7 ⊢ (𝑥 = 𝑤 → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) = if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤))))
100	95, 86	ifex 4536	. . . . . . 7 ⊢ if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤))) ∈ V
101	99, 10, 100	fvmpt 7013	. . . . . 6 ⊢ (𝑤 ∈ (0[,]1) → (𝐹‘𝑤) = if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤))))
102	97, 101	breqan12d 5122	. . . . 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 262	. . . 4 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1)) → (𝑧 < 𝑤 → (𝐹‘𝑧) < (𝐹‘𝑤)))
104	103	rgen2 3206	. . 3 ⊢ ∀𝑧 ∈ (0[,]1)∀𝑤 ∈ (0[,]1)(𝑧 < 𝑤 → (𝐹‘𝑧) < (𝐹‘𝑤))
105		soisoi 7356	. . 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 706	. 2 ⊢ 𝐹 Isom < , < ((0[,]1), (0[,]+∞))
107		letsr 18793	. . . . . 6 ⊢ ≤ ∈ TosetRel
108	107	elexi 3476	. . . . 5 ⊢ ≤ ∈ V
109	108	inex1 5281	. . . 4 ⊢ ( ≤ ∩ ((0[,]1) × (0[,]1))) ∈ V
110	108	inex1 5281	. . . 4 ⊢ ( ≤ ∩ ((0[,]+∞) × (0[,]+∞))) ∈ V
111		leiso 14628	. . . . . . . 8 ⊢ (((0[,]1) ⊆ ℝ^* ∧ (0[,]+∞) ⊆ ℝ^*) → (𝐹 Isom < , < ((0[,]1), (0[,]+∞)) ↔ 𝐹 Isom ≤ , ≤ ((0[,]1), (0[,]+∞))))
112	1, 5, 111	mp2an 705	. . . . . . 7 ⊢ (𝐹 Isom < , < ((0[,]1), (0[,]+∞)) ↔ 𝐹 Isom ≤ , ≤ ((0[,]1), (0[,]+∞)))
113	106, 112	mpbi 233	. . . . . 6 ⊢ 𝐹 Isom ≤ , ≤ ((0[,]1), (0[,]+∞))
114		isores1 7362	. . . . . 6 ⊢ (𝐹 Isom ≤ , ≤ ((0[,]1), (0[,]+∞)) ↔ 𝐹 Isom ( ≤ ∩ ((0[,]1) × (0[,]1))), ≤ ((0[,]1), (0[,]+∞)))
115	113, 114	mpbi 233	. . . . 5 ⊢ 𝐹 Isom ( ≤ ∩ ((0[,]1) × (0[,]1))), ≤ ((0[,]1), (0[,]+∞))
116		isores2 7361	. . . . 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 233	. . . 4 ⊢ 𝐹 Isom ( ≤ ∩ ((0[,]1) × (0[,]1))), ( ≤ ∩ ((0[,]+∞) × (0[,]+∞)))((0[,]1), (0[,]+∞))
118		tsrps 18787	. . . . . . . 8 ⊢ ( ≤ ∈ TosetRel → ≤ ∈ PosetRel)
119	107, 118	ax-mp 5	. . . . . . 7 ⊢ ≤ ∈ PosetRel
120		ledm 18790	. . . . . . . 8 ⊢ ℝ^* = dom ≤
121	120	psssdm 18782	. . . . . . 7 ⊢ (( ≤ ∈ PosetRel ∧ (0[,]1) ⊆ ℝ^*) → dom ( ≤ ∩ ((0[,]1) × (0[,]1))) = (0[,]1))
122	119, 1, 121	mp2an 705	. . . . . 6 ⊢ dom ( ≤ ∩ ((0[,]1) × (0[,]1))) = (0[,]1)
123	122	eqcomi 2772	. . . . 5 ⊢ (0[,]1) = dom ( ≤ ∩ ((0[,]1) × (0[,]1)))
124	120	psssdm 18782	. . . . . . 7 ⊢ (( ≤ ∈ PosetRel ∧ (0[,]+∞) ⊆ ℝ^*) → dom ( ≤ ∩ ((0[,]+∞) × (0[,]+∞))) = (0[,]+∞))
125	119, 5, 124	mp2an 705	. . . . . 6 ⊢ dom ( ≤ ∩ ((0[,]+∞) × (0[,]+∞))) = (0[,]+∞)
126	125	eqcomi 2772	. . . . 5 ⊢ (0[,]+∞) = dom ( ≤ ∩ ((0[,]+∞) × (0[,]+∞)))
127	123, 126	ordthmeo 24148	. . . 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 1491	. . 3 ⊢ 𝐹 ∈ ((ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1))))Homeo(ordTop‘( ≤ ∩ ((0[,]+∞) × (0[,]+∞)))))
129		dfii5 25233	. . . 4 ⊢ II = (ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1))))
130		iccpnfhmeo.k	. . . . 5 ⊢ 𝐾 = ((ordTop‘ ≤ ) ↾_t (0[,]+∞))
131		ordtresticc 23568	. . . . 5 ⊢ ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) = (ordTop‘( ≤ ∩ ((0[,]+∞) × (0[,]+∞))))
132	130, 131	eqtri 2786	. . . 4 ⊢ 𝐾 = (ordTop‘( ≤ ∩ ((0[,]+∞) × (0[,]+∞))))
133	129, 132	oveq12i 7452	. . 3 ⊢ (IIHomeo𝐾) = ((ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1))))Homeo(ordTop‘( ≤ ∩ ((0[,]+∞) × (0[,]+∞)))))
134	128, 133	eleqtrri 2863	. 2 ⊢ 𝐹 ∈ (IIHomeo𝐾)
135	106, 134	pm3.2i 476	1 ⊢ (𝐹 Isom < , < ((0[,]1), (0[,]+∞)) ∧ 𝐹 ∈ (IIHomeo𝐾))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∀wral 3080 Vcvv 3454 ⊆ wss 3900 ∩ cin 4092 ∪ cun 4152 ifcif 4485 {csn 4588 class class class wbr 5106 ↦ cmpt 5189 Po wpo 5561 Or wor 5562 × cxp 5653 ^◡ccnv 5654 dom cdm 5655 ⟶ontowfo 6553 ⟶1-1ontowf1o 6554 ‘cfv 6555 Isom wiso 6556 (class class class)co 7440 ℂcc 11222 ℝcr 11223 0cc0 11224 1c1 11225 + caddc 11227 +∞cpnf 11364 ℝ^*cxr 11366 < clt 11367 ≤ cle 11368 − cmin 11565 / cdiv 11997 [,)cico 13502 [,]cicc 13503 ↾_t crest 17617 TopOpenctopn 17618 ordTopcordt 17697 PosetRelcps 18764 TosetRel ctsr 18765 ℂ_fldccnfld 21705 Homeochmeo 24099 IIcii 25223

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-cnex 11280 ax-resscn 11281 ax-1cn 11282 ax-icn 11283 ax-addcl 11284 ax-addrcl 11285 ax-mulcl 11286 ax-mulrcl 11287 ax-mulcom 11288 ax-addass 11289 ax-mulass 11290 ax-distr 11291 ax-i2m1 11292 ax-1ne0 11293 ax-1rid 11294 ax-rnegex 11295 ax-rrecex 11296 ax-cnre 11297 ax-pre-lttri 11298 ax-pre-lttrn 11299 ax-pre-ltadd 11300 ax-pre-mulgt0 11301 ax-pre-sup 11302

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6319 df-ord 6380 df-on 6381 df-lim 6382 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-isom 6564 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-om 7899 df-1st 8022 df-2nd 8023 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-rdg 8443 df-1o 8501 df-2o 8502 df-er 8742 df-map 8874 df-en 8999 df-dom 9000 df-sdom 9001 df-fin 9002 df-fi 9427 df-sup 9458 df-inf 9459 df-pnf 11369 df-mnf 11370 df-xr 11371 df-ltxr 11372 df-le 11373 df-sub 11567 df-neg 11568 df-div 11998 df-nn 12360 df-2 12429 df-3 12430 df-4 12431 df-5 12432 df-6 12433 df-7 12434 df-8 12435 df-9 12436 df-n0 12631 df-z 12718 df-dec 12839 df-uz 12990 df-q 13100 df-rp 13145 df-xneg 13265 df-xadd 13266 df-xmul 13267 df-ioo 13504 df-ioc 13505 df-ico 13506 df-icc 13507 df-fz 13664 df-seq 14169 df-exp 14229 df-cj 15292 df-re 15293 df-im 15294 df-sqrt 15428 df-abs 15429 df-struct 17351 df-slot 17386 df-ndx 17398 df-base 17414 df-plusg 17467 df-mulr 17468 df-starv 17469 df-tset 17473 df-ple 17474 df-ds 17476 df-unif 17477 df-rest 17619 df-topn 17620 df-topgen 17640 df-ordt 17699 df-ps 18766 df-tsr 18767 df-psmet 21697 df-xmet 21698 df-met 21699 df-bl 21700 df-mopn 21701 df-cnfld 21706 df-top 23239 df-topon 23256 df-topsp 23278 df-bases 23291 df-cn 23572 df-hmeo 24101 df-xms 24666 df-ms 24667 df-ii 25225

This theorem is used by: xrhmeo 25294 xrge0hmph 34631

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