Theorem prdstmdd 24085

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 24036	. . . . 5 ⊢ (𝑥 ∈ TopMnd → 𝑥 ∈ Mnd)
6	5	ssriv 3939	. . . 4 ⊢ TopMnd ⊆ Mnd
7		fss 6688	. . . 4 ⊢ ((𝑅:𝐼⟶TopMnd ∧ TopMnd ⊆ Mnd) → 𝑅:𝐼⟶Mnd)
8	4, 6, 7	sylancl 587	. . 3 ⊢ (𝜑 → 𝑅:𝐼⟶Mnd)
9	1, 2, 3, 8	prdsmndd 18709	. 2 ⊢ (𝜑 → 𝑌 ∈ Mnd)
10		tmdtps 24037	. . . . 5 ⊢ (𝑥 ∈ TopMnd → 𝑥 ∈ TopSp)
11	10	ssriv 3939	. . . 4 ⊢ TopMnd ⊆ TopSp
12		fss 6688	. . . 4 ⊢ ((𝑅:𝐼⟶TopMnd ∧ TopMnd ⊆ TopSp) → 𝑅:𝐼⟶TopSp)
13	4, 11, 12	sylancl 587	. . 3 ⊢ (𝜑 → 𝑅:𝐼⟶TopSp)
14	1, 3, 2, 13	prdstps 23590	. 2 ⊢ (𝜑 → 𝑌 ∈ TopSp)
15		eqid 2737	. . . . . . 7 ⊢ (Base‘𝑌) = (Base‘𝑌)
16	3	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝑆 ∈ 𝑉)
17	2	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝐼 ∈ 𝑊)
18	4	ffnd 6673	. . . . . . . 8 ⊢ (𝜑 → 𝑅 Fn 𝐼)
19	18	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝑅 Fn 𝐼)
20		simp2 1138	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝑓 ∈ (Base‘𝑌))
21		simp3 1139	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝑔 ∈ (Base‘𝑌))
22		eqid 2737	. . . . . . 7 ⊢ (+g‘𝑌) = (+g‘𝑌)
23	1, 15, 16, 17, 19, 20, 21, 22	prdsplusgval 17407	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → (𝑓(+g‘𝑌)𝑔) = (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑘)(+g‘(𝑅‘𝑘))(𝑔‘𝑘))))
24	23	mpoeq3dva 7447	. . . . 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 2737	. . . . . 6 ⊢ (+𝑓‘𝑌) = (+𝑓‘𝑌)
26	15, 22, 25	plusffval 18585	. . . . 5 ⊢ (+𝑓‘𝑌) = (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑓(+g‘𝑌)𝑔))
27		vex 3446	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
28		vex 3446	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
29	27, 28	op1std 7955	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑓, 𝑔⟩ → (1st ‘𝑧) = 𝑓)
30	29	fveq1d 6846	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑓, 𝑔⟩ → ((1st ‘𝑧)‘𝑘) = (𝑓‘𝑘))
31	27, 28	op2ndd 7956	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑓, 𝑔⟩ → (2nd ‘𝑧) = 𝑔)
32	31	fveq1d 6846	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑓, 𝑔⟩ → ((2nd ‘𝑧)‘𝑘) = (𝑔‘𝑘))
33	30, 32	oveq12d 7388	. . . . . . 7 ⊢ (𝑧 = ⟨𝑓, 𝑔⟩ → (((1st ‘𝑧)‘𝑘)(+g‘(𝑅‘𝑘))((2nd ‘𝑧)‘𝑘)) = ((𝑓‘𝑘)(+g‘(𝑅‘𝑘))(𝑔‘𝑘)))
34	33	mpteq2dv 5194	. . . . . 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 7484	. . . . 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 2797	. . . 4 ⊢ (𝜑 → (+𝑓‘𝑌) = (𝑧 ∈ ((Base‘𝑌) × (Base‘𝑌)) ↦ (𝑘 ∈ 𝐼 ↦ (((1st ‘𝑧)‘𝑘)(+g‘(𝑅‘𝑘))((2nd ‘𝑧)‘𝑘)))))
37		eqid 2737	. . . . 5 ⊢ (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅))
38		eqid 2737	. . . . . . . 8 ⊢ (TopOpen‘𝑌) = (TopOpen‘𝑌)
39	15, 38	istps 22895	. . . . . . 7 ⊢ (𝑌 ∈ TopSp ↔ (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
40	14, 39	sylib 218	. . . . . 6 ⊢ (𝜑 → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
41		txtopon 23552	. . . . . 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 585	. . . . 5 ⊢ (𝜑 → ((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) ∈ (TopOn‘((Base‘𝑌) × (Base‘𝑌))))
43		topnfn 17359	. . . . . . . 8 ⊢ TopOpen Fn V
44		ssv 3960	. . . . . . . 8 ⊢ TopSp ⊆ V
45		fnssres 6625	. . . . . . . 8 ⊢ ((TopOpen Fn V ∧ TopSp ⊆ V) → (TopOpen ↾ TopSp) Fn TopSp)
46	43, 44, 45	mp2an 693	. . . . . . 7 ⊢ (TopOpen ↾ TopSp) Fn TopSp
47		fvres 6863	. . . . . . . . 9 ⊢ (𝑥 ∈ TopSp → ((TopOpen ↾ TopSp)‘𝑥) = (TopOpen‘𝑥))
48		eqid 2737	. . . . . . . . . 10 ⊢ (TopOpen‘𝑥) = (TopOpen‘𝑥)
49	48	tpstop 22898	. . . . . . . . 9 ⊢ (𝑥 ∈ TopSp → (TopOpen‘𝑥) ∈ Top)
50	47, 49	eqeltrd 2837	. . . . . . . 8 ⊢ (𝑥 ∈ TopSp → ((TopOpen ↾ TopSp)‘𝑥) ∈ Top)
51	50	rgen 3054	. . . . . . 7 ⊢ ∀𝑥 ∈ TopSp ((TopOpen ↾ TopSp)‘𝑥) ∈ Top
52		ffnfv 7075	. . . . . . 7 ⊢ ((TopOpen ↾ TopSp):TopSp⟶Top ↔ ((TopOpen ↾ TopSp) Fn TopSp ∧ ∀𝑥 ∈ TopSp ((TopOpen ↾ TopSp)‘𝑥) ∈ Top))
53	46, 51, 52	mpbir2an 712	. . . . . 6 ⊢ (TopOpen ↾ TopSp):TopSp⟶Top
54		fco2 6698	. . . . . 6 ⊢ (((TopOpen ↾ TopSp):TopSp⟶Top ∧ 𝑅:𝐼⟶TopSp) → (TopOpen ∘ 𝑅):𝐼⟶Top)
55	53, 13, 54	sylancr 588	. . . . 5 ⊢ (𝜑 → (TopOpen ∘ 𝑅):𝐼⟶Top)
56	33	mpompt 7484	. . . . . 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 2737	. . . . . . . 8 ⊢ (TopOpen‘(𝑅‘𝑘)) = (TopOpen‘(𝑅‘𝑘))
58		eqid 2737	. . . . . . . 8 ⊢ (+g‘(𝑅‘𝑘)) = (+g‘(𝑅‘𝑘))
59	4	ffvelcdmda 7040	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑅‘𝑘) ∈ TopMnd)
60	40	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
61	60, 60	cnmpt1st 23629	. . . . . . . . 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 23589	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
63	62	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
64	63, 60	eqeltrrd 2838	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)))
65		toponuni 22875	. . . . . . . . . . . . 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 5190	. . . . . . . . . . 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 2737	. . . . . . . . . . . . 13 ⊢ ∪ (∏t‘(TopOpen ∘ 𝑅)) = ∪ (∏t‘(TopOpen ∘ 𝑅))
72	71, 37	ptpjcn 23572	. . . . . . . . . . . 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 1374	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ ∪ (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥‘𝑘)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
74	67, 73	eqeltrd 2837	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥‘𝑘)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
75	63	eqcomd 2743	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (∏t‘(TopOpen ∘ 𝑅)) = (TopOpen‘𝑌))
76		fvco3 6943	. . . . . . . . . . . 12 ⊢ ((𝑅:𝐼⟶TopMnd ∧ 𝑘 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = (TopOpen‘(𝑅‘𝑘)))
77	4, 76	sylan 581	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = (TopOpen‘(𝑅‘𝑘)))
78	75, 77	oveq12d 7388	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)) = ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅‘𝑘))))
79	74, 78	eleqtrd 2839	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥‘𝑘)) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅‘𝑘))))
80		fveq1 6843	. . . . . . . . 9 ⊢ (𝑥 = 𝑓 → (𝑥‘𝑘) = (𝑓‘𝑘))
81	60, 60, 61, 60, 79, 80	cnmpt21 23632	. . . . . . . 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 23630	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ 𝑔) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)))
83		fveq1 6843	. . . . . . . . 9 ⊢ (𝑥 = 𝑔 → (𝑥‘𝑘) = (𝑔‘𝑘))
84	60, 60, 82, 60, 79, 83	cnmpt21 23632	. . . . . . . 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 24049	. . . . . . 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 7386	. . . . . . 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 2840	. . . . . 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 2841	. . . . 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 23588	. . . 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 2837	. . 3 ⊢ (𝜑 → (+𝑓‘𝑌) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (∏t‘(TopOpen ∘ 𝑅))))
91	62	oveq2d 7386	. . 3 ⊢ (𝜑 → (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)) = (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (∏t‘(TopOpen ∘ 𝑅))))
92	90, 91	eleqtrrd 2840	. 2 ⊢ (𝜑 → (+𝑓‘𝑌) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)))
93	25, 38	istmd 24035	. 2 ⊢ (𝑌 ∈ TopMnd ↔ (𝑌 ∈ Mnd ∧ 𝑌 ∈ TopSp ∧ (+𝑓‘𝑌) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌))))
94	9, 14, 92, 93	syl3anbrc 1345	1 ⊢ (𝜑 → 𝑌 ∈ TopMnd)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3442   ⊆ wss 3903  ⟨cop 4588  ∪ cuni 4865   ↦ cmpt 5181   × cxp 5632   ↾ cres 5636   ∘ ccom 5638   Fn wfn 6497  ⟶wf 6498  ‘cfv 6502  (class class class)co 7370   ∈ cmpo 7372  1^st c1st 7943  2^nd c2nd 7944  Basecbs 17150  +_gcplusg 17191  TopOpenctopn 17355  ∏_tcpt 17372  X_scprds 17379  +_𝑓cplusf 18576  Mndcmnd 18673  Topctop 22854  TopOnctopon 22871  TopSpctps 22893   Cn ccn 23185   ×_t ctx 23521  TopMndctmd 24031

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 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-cnex 11096  ax-resscn 11097  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-addrcl 11101  ax-mulcl 11102  ax-mulrcl 11103  ax-mulcom 11104  ax-addass 11105  ax-mulass 11106  ax-distr 11107  ax-i2m1 11108  ax-1ne0 11109  ax-1rid 11110  ax-rnegex 11111  ax-rrecex 11112  ax-cnre 11113  ax-pre-lttri 11114  ax-pre-lttrn 11115  ax-pre-ltadd 11116  ax-pre-mulgt0 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 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7821  df-1st 7945  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-1o 8409  df-2o 8410  df-er 8647  df-map 8779  df-ixp 8850  df-en 8898  df-dom 8899  df-sdom 8900  df-fin 8901  df-fi 9328  df-sup 9359  df-pnf 11182  df-mnf 11183  df-xr 11184  df-ltxr 11185  df-le 11186  df-sub 11380  df-neg 11381  df-nn 12160  df-2 12222  df-3 12223  df-4 12224  df-5 12225  df-6 12226  df-7 12227  df-8 12228  df-9 12229  df-n0 12416  df-z 12503  df-dec 12622  df-uz 12766  df-fz 13438  df-struct 17088  df-slot 17123  df-ndx 17135  df-base 17151  df-plusg 17204  df-mulr 17205  df-sca 17207  df-vsca 17208  df-ip 17209  df-tset 17210  df-ple 17211  df-ds 17213  df-hom 17215  df-cco 17216  df-rest 17356  df-topn 17357  df-0g 17375  df-topgen 17377  df-pt 17378  df-prds 17381  df-plusf 18578  df-mgm 18579  df-sgrp 18658  df-mnd 18674  df-top 22855  df-topon 22872  df-topsp 22894  df-bases 22907  df-cn 23188  df-cnp 23189  df-tx 23523  df-tmd 24033

This theorem is referenced by:  prdstgpd  24086