Theorem cvmlift3 35541

Description: A general version of cvmlift 35512. If 𝐾 is simply connected and weakly locally path-connected, then there is a unique lift of functions on 𝐾 which commutes with the covering map. (Contributed by Mario Carneiro, 9-Jul-2015.)

Hypotheses

Ref	Expression
cvmlift3.b	⊢ 𝐵 = ∪ 𝐶
cvmlift3.y	⊢ 𝑌 = ∪ 𝐾
cvmlift3.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
cvmlift3.k	⊢ (𝜑 → 𝐾 ∈ SConn)
cvmlift3.l	⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)
cvmlift3.o	⊢ (𝜑 → 𝑂 ∈ 𝑌)
cvmlift3.g	⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))
cvmlift3.p	⊢ (𝜑 → 𝑃 ∈ 𝐵)
cvmlift3.e	⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))

Assertion

Ref	Expression
cvmlift3	⊢ (𝜑 → ∃!𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃))

Distinct variable groups:   𝑓,𝐽   𝑓,𝐹   𝐵,𝑓   𝑓,𝐺   𝐶,𝑓   𝜑,𝑓   𝑓,𝐾   𝑃,𝑓   𝑓,𝑂   𝑓,𝑌

Proof of Theorem cvmlift3

Dummy variables 𝑏 𝑐 𝑑 𝑘 𝑠 𝑧 𝑔 𝑎 𝑢 𝑣 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvmlift3.b	. . 3 ⊢ 𝐵 = ∪ 𝐶
2		cvmlift3.y	. . 3 ⊢ 𝑌 = ∪ 𝐾
3		cvmlift3.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
4		cvmlift3.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ SConn)
5		cvmlift3.l	. . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)
6		cvmlift3.o	. . 3 ⊢ (𝜑 → 𝑂 ∈ 𝑌)
7		cvmlift3.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))
8		cvmlift3.p	. . 3 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
9		cvmlift3.e	. . 3 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))
10		eqeq2 2749	. . . . . . . 8 ⊢ (𝑏 = 𝑧 → (((℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑏 ↔ ((℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑧))
11	10	3anbi3d 1445	. . . . . . 7 ⊢ (𝑏 = 𝑧 → (((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑏) ↔ ((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑧)))
12	11	rexbidv 3162	. . . . . 6 ⊢ (𝑏 = 𝑧 → (∃𝑐 ∈ (II Cn 𝐾)((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑏) ↔ ∃𝑐 ∈ (II Cn 𝐾)((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑧)))
13	12	cbvriotavw 7335	. . . . 5 ⊢ (℩𝑏 ∈ 𝐵 ∃𝑐 ∈ (II Cn 𝐾)((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑏)) = (℩𝑧 ∈ 𝐵 ∃𝑐 ∈ (II Cn 𝐾)((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑧))
14		fveq1 6841	. . . . . . . . . 10 ⊢ (𝑐 = 𝑓 → (𝑐‘0) = (𝑓‘0))
15	14	eqeq1d 2739	. . . . . . . . 9 ⊢ (𝑐 = 𝑓 → ((𝑐‘0) = 𝑂 ↔ (𝑓‘0) = 𝑂))
16		fveq1 6841	. . . . . . . . . 10 ⊢ (𝑐 = 𝑓 → (𝑐‘1) = (𝑓‘1))
17	16	eqeq1d 2739	. . . . . . . . 9 ⊢ (𝑐 = 𝑓 → ((𝑐‘1) = 𝑎 ↔ (𝑓‘1) = 𝑎))
18		coeq2 5815	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑔 → (𝐹 ∘ 𝑑) = (𝐹 ∘ 𝑔))
19	18	eqeq1d 2739	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑔 → ((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑐)))
20		fveq1 6841	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑔 → (𝑑‘0) = (𝑔‘0))
21	20	eqeq1d 2739	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑔 → ((𝑑‘0) = 𝑃 ↔ (𝑔‘0) = 𝑃))
22	19, 21	anbi12d 633	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑔 → (((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑐) ∧ (𝑔‘0) = 𝑃)))
23	22	cbvriotavw 7335	. . . . . . . . . . . 12 ⊢ (℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃)) = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑐) ∧ (𝑔‘0) = 𝑃))
24		coeq2 5815	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝑓 → (𝐺 ∘ 𝑐) = (𝐺 ∘ 𝑓))
25	24	eqeq2d 2748	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑓 → ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑐) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓)))
26	25	anbi1d 632	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑓 → (((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑐) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃)))
27	26	riotabidv 7327	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑓 → (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑐) ∧ (𝑔‘0) = 𝑃)) = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃)))
28	23, 27	eqtrid 2784	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑓 → (℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃)) = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃)))
29	28	fveq1d 6844	. . . . . . . . . 10 ⊢ (𝑐 = 𝑓 → ((℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1))
30	29	eqeq1d 2739	. . . . . . . . 9 ⊢ (𝑐 = 𝑓 → (((℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑧 ↔ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧))
31	15, 17, 30	3anbi123d 1439	. . . . . . . 8 ⊢ (𝑐 = 𝑓 → (((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑧) ↔ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑎 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
32	31	cbvrexvw 3217	. . . . . . 7 ⊢ (∃𝑐 ∈ (II Cn 𝐾)((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑧) ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑎 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧))
33		eqeq2 2749	. . . . . . . . 9 ⊢ (𝑎 = 𝑥 → ((𝑓‘1) = 𝑎 ↔ (𝑓‘1) = 𝑥))
34	33	3anbi2d 1444	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑎 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧) ↔ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
35	34	rexbidv 3162	. . . . . . 7 ⊢ (𝑎 = 𝑥 → (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑎 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧) ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
36	32, 35	bitrid 283	. . . . . 6 ⊢ (𝑎 = 𝑥 → (∃𝑐 ∈ (II Cn 𝐾)((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑧) ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
37	36	riotabidv 7327	. . . . 5 ⊢ (𝑎 = 𝑥 → (℩𝑧 ∈ 𝐵 ∃𝑐 ∈ (II Cn 𝐾)((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑧)) = (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
38	13, 37	eqtrid 2784	. . . 4 ⊢ (𝑎 = 𝑥 → (℩𝑏 ∈ 𝐵 ∃𝑐 ∈ (II Cn 𝐾)((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑏)) = (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
39	38	cbvmptv 5204	. . 3 ⊢ (𝑎 ∈ 𝑌 ↦ (℩𝑏 ∈ 𝐵 ∃𝑐 ∈ (II Cn 𝐾)((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑏))) = (𝑥 ∈ 𝑌 ↦ (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
40		eqid 2737	. . . 4 ⊢ (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))}) = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))})
41	40	cvmscbv 35471	. . 3 ⊢ (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))}) = (𝑎 ∈ 𝐽 ↦ {𝑏 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑏 = (◡𝐹 “ 𝑎) ∧ ∀𝑣 ∈ 𝑏 (∀𝑢 ∈ (𝑏 ∖ {𝑣})(𝑣 ∩ 𝑢) = ∅ ∧ (𝐹 ↾ 𝑣) ∈ ((𝐶 ↾t 𝑣)Homeo(𝐽 ↾t 𝑎))))})
42	1, 2, 3, 4, 5, 6, 7, 8, 9, 39, 41	cvmlift3lem9 35540	. 2 ⊢ (𝜑 → ∃𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃))
43		sconnpconn 35440	. . . 4 ⊢ (𝐾 ∈ SConn → 𝐾 ∈ PConn)
44		pconnconn 35444	. . . 4 ⊢ (𝐾 ∈ PConn → 𝐾 ∈ Conn)
45	4, 43, 44	3syl 18	. . 3 ⊢ (𝜑 → 𝐾 ∈ Conn)
46		pconnconn 35444	. . . . . 6 ⊢ (𝑥 ∈ PConn → 𝑥 ∈ Conn)
47	46	ssriv 3939	. . . . 5 ⊢ PConn ⊆ Conn
48		nllyss 23436	. . . . 5 ⊢ (PConn ⊆ Conn → 𝑛-Locally PConn ⊆ 𝑛-Locally Conn)
49	47, 48	ax-mp 5	. . . 4 ⊢ 𝑛-Locally PConn ⊆ 𝑛-Locally Conn
50	49, 5	sselid 3933	. . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Conn)
51	1, 2, 3, 45, 50, 6, 7, 8, 9	cvmliftmo 35497	. 2 ⊢ (𝜑 → ∃*𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃))
52		reu5 3354	. 2 ⊢ (∃!𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃) ↔ (∃𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃) ∧ ∃*𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃)))
53	42, 51, 52	sylanbrc 584	1 ⊢ (𝜑 → ∃!𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  ∃!wreu 3350  ∃*wrmo 3351  {crab 3401   ∖ cdif 3900   ∩ cin 3902   ⊆ wss 3903  ∅c0 4287  𝒫 cpw 4556  {csn 4582  ∪ cuni 4865   ↦ cmpt 5181  ^◡ccnv 5631   ↾ cres 5634   “ cima 5635   ∘ ccom 5636  ‘cfv 6500  ℩crio 7324  (class class class)co 7368  0cc0 11038  1c1 11039   ↾_t crest 17352   Cn ccn 23180  Conncconn 23367  𝑛-Locally cnlly 23421  Homeochmeo 23709  IIcii 24836  PConncpconn 35432  SConncsconn 35433   CovMap ccvm 35468

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-inf2 9562  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

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-ec 8647  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-ico 13279  df-icc 13280  df-fz 13436  df-fzo 13583  df-fl 13724  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-clim 15423  df-sum 15622  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-cld 22975  df-ntr 22976  df-cls 22977  df-nei 23054  df-cn 23183  df-cnp 23184  df-cmp 23343  df-conn 23368  df-lly 23422  df-nlly 23423  df-tx 23518  df-hmeo 23711  df-xms 24276  df-ms 24277  df-tms 24278  df-ii 24838  df-cncf 24839  df-htpy 24937  df-phtpy 24938  df-phtpc 24959  df-pco 24973  df-pconn 35434  df-sconn 35435  df-cvm 35469

This theorem is referenced by: (None)