Theorem prdstmdd 24062

Description: The product of a family of topological monoids is a topological monoid. (Contributed by Mario Carneiro, 22-Sep-2015.)

Hypotheses

Ref	Expression
prdstmdd.y	⊢ 𝑌 = (𝑆Xs𝑅)
prdstmdd.i	⊢ (𝜑 → 𝐼 ∈ 𝑊)
prdstmdd.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)
prdstmdd.r	⊢ (𝜑 → 𝑅:𝐼⟶TopMnd)

Assertion

Ref	Expression
prdstmdd	⊢ (𝜑 → 𝑌 ∈ TopMnd)

Proof of Theorem prdstmdd

Dummy variables 𝑓 𝑔 𝑘 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prdstmdd.y	. . 3 ⊢ 𝑌 = (𝑆Xs𝑅)
2		prdstmdd.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
3		prdstmdd.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
4		prdstmdd.r	. . . 4 ⊢ (𝜑 → 𝑅:𝐼⟶TopMnd)
5		tmdmnd 24013	. . . . 5 ⊢ (𝑥 ∈ TopMnd → 𝑥 ∈ Mnd)
6	5	ssriv 3962	. . . 4 ⊢ TopMnd ⊆ Mnd
7		fss 6722	. . . 4 ⊢ ((𝑅:𝐼⟶TopMnd ∧ TopMnd ⊆ Mnd) → 𝑅:𝐼⟶Mnd)
8	4, 6, 7	sylancl 586	. . 3 ⊢ (𝜑 → 𝑅:𝐼⟶Mnd)
9	1, 2, 3, 8	prdsmndd 18748	. 2 ⊢ (𝜑 → 𝑌 ∈ Mnd)
10		tmdtps 24014	. . . . 5 ⊢ (𝑥 ∈ TopMnd → 𝑥 ∈ TopSp)
11	10	ssriv 3962	. . . 4 ⊢ TopMnd ⊆ TopSp
12		fss 6722	. . . 4 ⊢ ((𝑅:𝐼⟶TopMnd ∧ TopMnd ⊆ TopSp) → 𝑅:𝐼⟶TopSp)
13	4, 11, 12	sylancl 586	. . 3 ⊢ (𝜑 → 𝑅:𝐼⟶TopSp)
14	1, 3, 2, 13	prdstps 23567	. 2 ⊢ (𝜑 → 𝑌 ∈ TopSp)
15		eqid 2735	. . . . . . 7 ⊢ (Base‘𝑌) = (Base‘𝑌)
16	3	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝑆 ∈ 𝑉)
17	2	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝐼 ∈ 𝑊)
18	4	ffnd 6707	. . . . . . . 8 ⊢ (𝜑 → 𝑅 Fn 𝐼)
19	18	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝑅 Fn 𝐼)
20		simp2 1137	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝑓 ∈ (Base‘𝑌))
21		simp3 1138	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝑔 ∈ (Base‘𝑌))
22		eqid 2735	. . . . . . 7 ⊢ (+g‘𝑌) = (+g‘𝑌)
23	1, 15, 16, 17, 19, 20, 21, 22	prdsplusgval 17487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → (𝑓(+g‘𝑌)𝑔) = (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑘)(+g‘(𝑅‘𝑘))(𝑔‘𝑘))))
24	23	mpoeq3dva 7484	. . . . 5 ⊢ (𝜑 → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑓(+g‘𝑌)𝑔)) = (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑘)(+g‘(𝑅‘𝑘))(𝑔‘𝑘)))))
25		eqid 2735	. . . . . 6 ⊢ (+𝑓‘𝑌) = (+𝑓‘𝑌)
26	15, 22, 25	plusffval 18624	. . . . 5 ⊢ (+𝑓‘𝑌) = (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑓(+g‘𝑌)𝑔))
27		vex 3463	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
28		vex 3463	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
29	27, 28	op1std 7998	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑓, 𝑔⟩ → (1st ‘𝑧) = 𝑓)
30	29	fveq1d 6878	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑓, 𝑔⟩ → ((1st ‘𝑧)‘𝑘) = (𝑓‘𝑘))
31	27, 28	op2ndd 7999	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑓, 𝑔⟩ → (2nd ‘𝑧) = 𝑔)
32	31	fveq1d 6878	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑓, 𝑔⟩ → ((2nd ‘𝑧)‘𝑘) = (𝑔‘𝑘))
33	30, 32	oveq12d 7423	. . . . . . 7 ⊢ (𝑧 = ⟨𝑓, 𝑔⟩ → (((1st ‘𝑧)‘𝑘)(+g‘(𝑅‘𝑘))((2nd ‘𝑧)‘𝑘)) = ((𝑓‘𝑘)(+g‘(𝑅‘𝑘))(𝑔‘𝑘)))
34	33	mpteq2dv 5215	. . . . . 6 ⊢ (𝑧 = ⟨𝑓, 𝑔⟩ → (𝑘 ∈ 𝐼 ↦ (((1st ‘𝑧)‘𝑘)(+g‘(𝑅‘𝑘))((2nd ‘𝑧)‘𝑘))) = (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑘)(+g‘(𝑅‘𝑘))(𝑔‘𝑘))))
35	34	mpompt 7521	. . . . 5 ⊢ (𝑧 ∈ ((Base‘𝑌) × (Base‘𝑌)) ↦ (𝑘 ∈ 𝐼 ↦ (((1st ‘𝑧)‘𝑘)(+g‘(𝑅‘𝑘))((2nd ‘𝑧)‘𝑘)))) = (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑘)(+g‘(𝑅‘𝑘))(𝑔‘𝑘))))
36	24, 26, 35	3eqtr4g 2795	. . . 4 ⊢ (𝜑 → (+𝑓‘𝑌) = (𝑧 ∈ ((Base‘𝑌) × (Base‘𝑌)) ↦ (𝑘 ∈ 𝐼 ↦ (((1st ‘𝑧)‘𝑘)(+g‘(𝑅‘𝑘))((2nd ‘𝑧)‘𝑘)))))
37		eqid 2735	. . . . 5 ⊢ (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅))
38		eqid 2735	. . . . . . . 8 ⊢ (TopOpen‘𝑌) = (TopOpen‘𝑌)
39	15, 38	istps 22872	. . . . . . 7 ⊢ (𝑌 ∈ TopSp ↔ (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
40	14, 39	sylib 218	. . . . . 6 ⊢ (𝜑 → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
41		txtopon 23529	. . . . . 6 ⊢ (((TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)) ∧ (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌))) → ((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) ∈ (TopOn‘((Base‘𝑌) × (Base‘𝑌))))
42	40, 40, 41	syl2anc 584	. . . . 5 ⊢ (𝜑 → ((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) ∈ (TopOn‘((Base‘𝑌) × (Base‘𝑌))))
43		topnfn 17439	. . . . . . . 8 ⊢ TopOpen Fn V
44		ssv 3983	. . . . . . . 8 ⊢ TopSp ⊆ V
45		fnssres 6661	. . . . . . . 8 ⊢ ((TopOpen Fn V ∧ TopSp ⊆ V) → (TopOpen ↾ TopSp) Fn TopSp)
46	43, 44, 45	mp2an 692	. . . . . . 7 ⊢ (TopOpen ↾ TopSp) Fn TopSp
47		fvres 6895	. . . . . . . . 9 ⊢ (𝑥 ∈ TopSp → ((TopOpen ↾ TopSp)‘𝑥) = (TopOpen‘𝑥))
48		eqid 2735	. . . . . . . . . 10 ⊢ (TopOpen‘𝑥) = (TopOpen‘𝑥)
49	48	tpstop 22875	. . . . . . . . 9 ⊢ (𝑥 ∈ TopSp → (TopOpen‘𝑥) ∈ Top)
50	47, 49	eqeltrd 2834	. . . . . . . 8 ⊢ (𝑥 ∈ TopSp → ((TopOpen ↾ TopSp)‘𝑥) ∈ Top)
51	50	rgen 3053	. . . . . . 7 ⊢ ∀𝑥 ∈ TopSp ((TopOpen ↾ TopSp)‘𝑥) ∈ Top
52		ffnfv 7109	. . . . . . 7 ⊢ ((TopOpen ↾ TopSp):TopSp⟶Top ↔ ((TopOpen ↾ TopSp) Fn TopSp ∧ ∀𝑥 ∈ TopSp ((TopOpen ↾ TopSp)‘𝑥) ∈ Top))
53	46, 51, 52	mpbir2an 711	. . . . . 6 ⊢ (TopOpen ↾ TopSp):TopSp⟶Top
54		fco2 6732	. . . . . 6 ⊢ (((TopOpen ↾ TopSp):TopSp⟶Top ∧ 𝑅:𝐼⟶TopSp) → (TopOpen ∘ 𝑅):𝐼⟶Top)
55	53, 13, 54	sylancr 587	. . . . 5 ⊢ (𝜑 → (TopOpen ∘ 𝑅):𝐼⟶Top)
56	33	mpompt 7521	. . . . . 6 ⊢ (𝑧 ∈ ((Base‘𝑌) × (Base‘𝑌)) ↦ (((1st ‘𝑧)‘𝑘)(+g‘(𝑅‘𝑘))((2nd ‘𝑧)‘𝑘))) = (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ ((𝑓‘𝑘)(+g‘(𝑅‘𝑘))(𝑔‘𝑘)))
57		eqid 2735	. . . . . . . 8 ⊢ (TopOpen‘(𝑅‘𝑘)) = (TopOpen‘(𝑅‘𝑘))
58		eqid 2735	. . . . . . . 8 ⊢ (+g‘(𝑅‘𝑘)) = (+g‘(𝑅‘𝑘))
59	4	ffvelcdmda 7074	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑅‘𝑘) ∈ TopMnd)
60	40	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
61	60, 60	cnmpt1st 23606	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ 𝑓) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)))
62	1, 3, 2, 18, 38	prdstopn 23566	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
63	62	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
64	63, 60	eqeltrrd 2835	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)))
65		toponuni 22852	. . . . . . . . . . . . 13 ⊢ ((∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)) → (Base‘𝑌) = ∪ (∏t‘(TopOpen ∘ 𝑅)))
66	64, 65	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (Base‘𝑌) = ∪ (∏t‘(TopOpen ∘ 𝑅)))
67	66	mpteq1d 5210	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥‘𝑘)) = (𝑥 ∈ ∪ (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥‘𝑘)))
68	2	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐼 ∈ 𝑊)
69	55	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (TopOpen ∘ 𝑅):𝐼⟶Top)
70		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝑘 ∈ 𝐼)
71		eqid 2735	. . . . . . . . . . . . 13 ⊢ ∪ (∏t‘(TopOpen ∘ 𝑅)) = ∪ (∏t‘(TopOpen ∘ 𝑅))
72	71, 37	ptpjcn 23549	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑊 ∧ (TopOpen ∘ 𝑅):𝐼⟶Top ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ ∪ (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥‘𝑘)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
73	68, 69, 70, 72	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ ∪ (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥‘𝑘)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
74	67, 73	eqeltrd 2834	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥‘𝑘)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
75	63	eqcomd 2741	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (∏t‘(TopOpen ∘ 𝑅)) = (TopOpen‘𝑌))
76		fvco3 6978	. . . . . . . . . . . 12 ⊢ ((𝑅:𝐼⟶TopMnd ∧ 𝑘 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = (TopOpen‘(𝑅‘𝑘)))
77	4, 76	sylan 580	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = (TopOpen‘(𝑅‘𝑘)))
78	75, 77	oveq12d 7423	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)) = ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅‘𝑘))))
79	74, 78	eleqtrd 2836	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥‘𝑘)) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅‘𝑘))))
80		fveq1 6875	. . . . . . . . 9 ⊢ (𝑥 = 𝑓 → (𝑥‘𝑘) = (𝑓‘𝑘))
81	60, 60, 61, 60, 79, 80	cnmpt21 23609	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑓‘𝑘)) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘(𝑅‘𝑘))))
82	60, 60	cnmpt2nd 23607	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ 𝑔) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)))
83		fveq1 6875	. . . . . . . . 9 ⊢ (𝑥 = 𝑔 → (𝑥‘𝑘) = (𝑔‘𝑘))
84	60, 60, 82, 60, 79, 83	cnmpt21 23609	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑔‘𝑘)) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘(𝑅‘𝑘))))
85	57, 58, 59, 60, 60, 81, 84	cnmpt2plusg 24026	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ ((𝑓‘𝑘)(+g‘(𝑅‘𝑘))(𝑔‘𝑘))) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘(𝑅‘𝑘))))
86	77	oveq2d 7421	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn ((TopOpen ∘ 𝑅)‘𝑘)) = (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘(𝑅‘𝑘))))
87	85, 86	eleqtrrd 2837	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ ((𝑓‘𝑘)(+g‘(𝑅‘𝑘))(𝑔‘𝑘))) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
88	56, 87	eqeltrid 2838	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑧 ∈ ((Base‘𝑌) × (Base‘𝑌)) ↦ (((1st ‘𝑧)‘𝑘)(+g‘(𝑅‘𝑘))((2nd ‘𝑧)‘𝑘))) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
89	37, 42, 2, 55, 88	ptcn 23565	. . . 4 ⊢ (𝜑 → (𝑧 ∈ ((Base‘𝑌) × (Base‘𝑌)) ↦ (𝑘 ∈ 𝐼 ↦ (((1st ‘𝑧)‘𝑘)(+g‘(𝑅‘𝑘))((2nd ‘𝑧)‘𝑘)))) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (∏t‘(TopOpen ∘ 𝑅))))
90	36, 89	eqeltrd 2834	. . 3 ⊢ (𝜑 → (+𝑓‘𝑌) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (∏t‘(TopOpen ∘ 𝑅))))
91	62	oveq2d 7421	. . 3 ⊢ (𝜑 → (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)) = (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (∏t‘(TopOpen ∘ 𝑅))))
92	90, 91	eleqtrrd 2837	. 2 ⊢ (𝜑 → (+𝑓‘𝑌) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)))
93	25, 38	istmd 24012	. 2 ⊢ (𝑌 ∈ TopMnd ↔ (𝑌 ∈ Mnd ∧ 𝑌 ∈ TopSp ∧ (+𝑓‘𝑌) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌))))
94	9, 14, 92, 93	syl3anbrc 1344	1 ⊢ (𝜑 → 𝑌 ∈ TopMnd)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ∀wral 3051  Vcvv 3459   ⊆ wss 3926  ⟨cop 4607  ∪ cuni 4883   ↦ cmpt 5201   × cxp 5652   ↾ cres 5656   ∘ ccom 5658   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  (class class class)co 7405   ∈ cmpo 7407  1^st c1st 7986  2^nd c2nd 7987  Basecbs 17228  +_gcplusg 17271  TopOpenctopn 17435  ∏_tcpt 17452  X_scprds 17459  +_𝑓cplusf 18615  Mndcmnd 18712  Topctop 22831  TopOnctopon 22848  TopSpctps 22870   Cn ccn 23162   ×_t ctx 23498  TopMndctmd 24008

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-iin 4970  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-er 8719  df-map 8842  df-ixp 8912  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-fi 9423  df-sup 9454  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12502  df-z 12589  df-dec 12709  df-uz 12853  df-fz 13525  df-struct 17166  df-slot 17201  df-ndx 17213  df-base 17229  df-plusg 17284  df-mulr 17285  df-sca 17287  df-vsca 17288  df-ip 17289  df-tset 17290  df-ple 17291  df-ds 17293  df-hom 17295  df-cco 17296  df-rest 17436  df-topn 17437  df-0g 17455  df-topgen 17457  df-pt 17458  df-prds 17461  df-plusf 18617  df-mgm 18618  df-sgrp 18697  df-mnd 18713  df-top 22832  df-topon 22849  df-topsp 22871  df-bases 22884  df-cn 23165  df-cnp 23166  df-tx 23500  df-tmd 24010

This theorem is referenced by:  prdstgpd  24063