Theorem cvmlift3 33286

Description: A general version of cvmlift 33257. 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 2752	. . . . . . . 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 1441	. . . . . . 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 3228	. . . . . 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 7238	. . . . 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 6770	. . . . . . . . . 10 ⊢ (𝑐 = 𝑓 → (𝑐‘0) = (𝑓‘0))
15	14	eqeq1d 2742	. . . . . . . . 9 ⊢ (𝑐 = 𝑓 → ((𝑐‘0) = 𝑂 ↔ (𝑓‘0) = 𝑂))
16		fveq1 6770	. . . . . . . . . 10 ⊢ (𝑐 = 𝑓 → (𝑐‘1) = (𝑓‘1))
17	16	eqeq1d 2742	. . . . . . . . 9 ⊢ (𝑐 = 𝑓 → ((𝑐‘1) = 𝑎 ↔ (𝑓‘1) = 𝑎))
18		coeq2 5766	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑔 → (𝐹 ∘ 𝑑) = (𝐹 ∘ 𝑔))
19	18	eqeq1d 2742	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑔 → ((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑐)))
20		fveq1 6770	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑔 → (𝑑‘0) = (𝑔‘0))
21	20	eqeq1d 2742	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑔 → ((𝑑‘0) = 𝑃 ↔ (𝑔‘0) = 𝑃))
22	19, 21	anbi12d 631	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑔 → (((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑐) ∧ (𝑔‘0) = 𝑃)))
23	22	cbvriotavw 7238	. . . . . . . . . . . 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 5766	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝑓 → (𝐺 ∘ 𝑐) = (𝐺 ∘ 𝑓))
25	24	eqeq2d 2751	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑓 → ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑐) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓)))
26	25	anbi1d 630	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑓 → (((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑐) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃)))
27	26	riotabidv 7230	. . . . . . . . . . . 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 2792	. . . . . . . . . . 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 6773	. . . . . . . . . 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 2742	. . . . . . . . 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 1435	. . . . . . . 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 3382	. . . . . . 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 2752	. . . . . . . . 9 ⊢ (𝑎 = 𝑥 → ((𝑓‘1) = 𝑎 ↔ (𝑓‘1) = 𝑥))
34	33	3anbi2d 1440	. . . . . . . 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 3228	. . . . . . 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	syl5bb 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 7230	. . . . 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 2792	. . . 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 5192	. . 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 2740	. . . 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 33216	. . 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 33285	. 2 ⊢ (𝜑 → ∃𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃))
43		sconnpconn 33185	. . . 4 ⊢ (𝐾 ∈ SConn → 𝐾 ∈ PConn)
44		pconnconn 33189	. . . 4 ⊢ (𝐾 ∈ PConn → 𝐾 ∈ Conn)
45	4, 43, 44	3syl 18	. . 3 ⊢ (𝜑 → 𝐾 ∈ Conn)
46		pconnconn 33189	. . . . . 6 ⊢ (𝑥 ∈ PConn → 𝑥 ∈ Conn)
47	46	ssriv 3930	. . . . 5 ⊢ PConn ⊆ Conn
48		nllyss 22629	. . . . 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 3924	. . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Conn)
51	1, 2, 3, 45, 50, 6, 7, 8, 9	cvmliftmo 33242	. 2 ⊢ (𝜑 → ∃*𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃))
52		reu5 3360	. 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 583	1 ⊢ (𝜑 → ∃!𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1542   ∈ wcel 2110  ∀wral 3066  ∃wrex 3067  ∃!wreu 3068  ∃*wrmo 3069  {crab 3070   ∖ cdif 3889   ∩ cin 3891   ⊆ wss 3892  ∅c0 4262  𝒫 cpw 4539  {csn 4567  ∪ cuni 4845   ↦ cmpt 5162  ^◡ccnv 5589   ↾ cres 5592   “ cima 5593   ∘ ccom 5594  ‘cfv 6432  ℩crio 7227  (class class class)co 7271  0cc0 10872  1c1 10873   ↾_t crest 17129   Cn ccn 22373  Conncconn 22560  𝑛-Locally cnlly 22614  Homeochmeo 22902  IIcii 24036  PConncpconn 33177  SConncsconn 33178   CovMap ccvm 33213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582  ax-inf2 9377  ax-cnex 10928  ax-resscn 10929  ax-1cn 10930  ax-icn 10931  ax-addcl 10932  ax-addrcl 10933  ax-mulcl 10934  ax-mulrcl 10935  ax-mulcom 10936  ax-addass 10937  ax-mulass 10938  ax-distr 10939  ax-i2m1 10940  ax-1ne0 10941  ax-1rid 10942  ax-rnegex 10943  ax-rrecex 10944  ax-cnre 10945  ax-pre-lttri 10946  ax-pre-lttrn 10947  ax-pre-ltadd 10948  ax-pre-mulgt0 10949  ax-pre-sup 10950  ax-addf 10951  ax-mulf 10952

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-iin 4933  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-se 5546  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-isom 6441  df-riota 7228  df-ov 7274  df-oprab 7275  df-mpo 7276  df-of 7527  df-om 7707  df-1st 7824  df-2nd 7825  df-supp 7969  df-frecs 8088  df-wrecs 8119  df-recs 8193  df-rdg 8232  df-1o 8288  df-2o 8289  df-er 8481  df-ec 8483  df-map 8600  df-ixp 8669  df-en 8717  df-dom 8718  df-sdom 8719  df-fin 8720  df-fsupp 9107  df-fi 9148  df-sup 9179  df-inf 9180  df-oi 9247  df-card 9698  df-pnf 11012  df-mnf 11013  df-xr 11014  df-ltxr 11015  df-le 11016  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12437  df-uz 12582  df-q 12688  df-rp 12730  df-xneg 12847  df-xadd 12848  df-xmul 12849  df-ioo 13082  df-ico 13084  df-icc 13085  df-fz 13239  df-fzo 13382  df-fl 13510  df-seq 13720  df-exp 13781  df-hash 14043  df-cj 14808  df-re 14809  df-im 14810  df-sqrt 14944  df-abs 14945  df-clim 15195  df-sum 15396  df-struct 16846  df-sets 16863  df-slot 16881  df-ndx 16893  df-base 16911  df-ress 16940  df-plusg 16973  df-mulr 16974  df-starv 16975  df-sca 16976  df-vsca 16977  df-ip 16978  df-tset 16979  df-ple 16980  df-ds 16982  df-unif 16983  df-hom 16984  df-cco 16985  df-rest 17131  df-topn 17132  df-0g 17150  df-gsum 17151  df-topgen 17152  df-pt 17153  df-prds 17156  df-xrs 17211  df-qtop 17216  df-imas 17217  df-xps 17219  df-mre 17293  df-mrc 17294  df-acs 17296  df-mgm 18324  df-sgrp 18373  df-mnd 18384  df-submnd 18429  df-mulg 18699  df-cntz 18921  df-cmn 19386  df-psmet 20587  df-xmet 20588  df-met 20589  df-bl 20590  df-mopn 20591  df-cnfld 20596  df-top 22041  df-topon 22058  df-topsp 22080  df-bases 22094  df-cld 22168  df-ntr 22169  df-cls 22170  df-nei 22247  df-cn 22376  df-cnp 22377  df-cmp 22536  df-conn 22561  df-lly 22615  df-nlly 22616  df-tx 22711  df-hmeo 22904  df-xms 23471  df-ms 23472  df-tms 23473  df-ii 24038  df-htpy 24131  df-phtpy 24132  df-phtpc 24153  df-pco 24166  df-pconn 33179  df-sconn 33180  df-cvm 33214

This theorem is referenced by: (None)