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Theorem prdstmdd 23183

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 = (𝑆X_s𝑅)
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 = (𝑆X_s𝑅)
2		prdstmdd.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
3		prdstmdd.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
4		prdstmdd.r	. . . 4 ⊢ (𝜑 → 𝑅:𝐼⟶TopMnd)
5		tmdmnd 23134	. . . . 5 ⊢ (𝑥 ∈ TopMnd → 𝑥 ∈ Mnd)
6	5	ssriv 3921	. . . 4 ⊢ TopMnd ⊆ Mnd
7		fss 6601	. . . 4 ⊢ ((𝑅:𝐼⟶TopMnd ∧ TopMnd ⊆ Mnd) → 𝑅:𝐼⟶Mnd)
8	4, 6, 7	sylancl 585	. . 3 ⊢ (𝜑 → 𝑅:𝐼⟶Mnd)
9	1, 2, 3, 8	prdsmndd 18333	. 2 ⊢ (𝜑 → 𝑌 ∈ Mnd)
10		tmdtps 23135	. . . . 5 ⊢ (𝑥 ∈ TopMnd → 𝑥 ∈ TopSp)
11	10	ssriv 3921	. . . 4 ⊢ TopMnd ⊆ TopSp
12		fss 6601	. . . 4 ⊢ ((𝑅:𝐼⟶TopMnd ∧ TopMnd ⊆ TopSp) → 𝑅:𝐼⟶TopSp)
13	4, 11, 12	sylancl 585	. . 3 ⊢ (𝜑 → 𝑅:𝐼⟶TopSp)
14	1, 3, 2, 13	prdstps 22688	. 2 ⊢ (𝜑 → 𝑌 ∈ TopSp)
15		eqid 2738	. . . . . . 7 ⊢ (Base‘𝑌) = (Base‘𝑌)
16	3	3ad2ant1 1131	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝑆 ∈ 𝑉)
17	2	3ad2ant1 1131	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝐼 ∈ 𝑊)
18	4	ffnd 6585	. . . . . . . 8 ⊢ (𝜑 → 𝑅 Fn 𝐼)
19	18	3ad2ant1 1131	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝑅 Fn 𝐼)
20		simp2 1135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝑓 ∈ (Base‘𝑌))
21		simp3 1136	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝑔 ∈ (Base‘𝑌))
22		eqid 2738	. . . . . . 7 ⊢ (+_g‘𝑌) = (+_g‘𝑌)
23	1, 15, 16, 17, 19, 20, 21, 22	prdsplusgval 17101	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → (𝑓(+_g‘𝑌)𝑔) = (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑘)(+_g‘(𝑅‘𝑘))(𝑔‘𝑘))))
24	23	mpoeq3dva 7330	. . . . 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 2738	. . . . . 6 ⊢ (+_𝑓‘𝑌) = (+_𝑓‘𝑌)
26	15, 22, 25	plusffval 18247	. . . . 5 ⊢ (+_𝑓‘𝑌) = (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑓(+_g‘𝑌)𝑔))
27		vex 3426	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
28		vex 3426	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
29	27, 28	op1std 7814	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑓, 𝑔⟩ → (1^st ‘𝑧) = 𝑓)
30	29	fveq1d 6758	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑓, 𝑔⟩ → ((1^st ‘𝑧)‘𝑘) = (𝑓‘𝑘))
31	27, 28	op2ndd 7815	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑓, 𝑔⟩ → (2^nd ‘𝑧) = 𝑔)
32	31	fveq1d 6758	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑓, 𝑔⟩ → ((2^nd ‘𝑧)‘𝑘) = (𝑔‘𝑘))
33	30, 32	oveq12d 7273	. . . . . . 7 ⊢ (𝑧 = ⟨𝑓, 𝑔⟩ → (((1^st ‘𝑧)‘𝑘)(+_g‘(𝑅‘𝑘))((2^nd ‘𝑧)‘𝑘)) = ((𝑓‘𝑘)(+_g‘(𝑅‘𝑘))(𝑔‘𝑘)))
34	33	mpteq2dv 5172	. . . . . 6 ⊢ (𝑧 = ⟨𝑓, 𝑔⟩ → (𝑘 ∈ 𝐼 ↦ (((1^st ‘𝑧)‘𝑘)(+_g‘(𝑅‘𝑘))((2^nd ‘𝑧)‘𝑘))) = (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑘)(+_g‘(𝑅‘𝑘))(𝑔‘𝑘))))
35	34	mpompt 7366	. . . . 5 ⊢ (𝑧 ∈ ((Base‘𝑌) × (Base‘𝑌)) ↦ (𝑘 ∈ 𝐼 ↦ (((1^st ‘𝑧)‘𝑘)(+_g‘(𝑅‘𝑘))((2^nd ‘𝑧)‘𝑘)))) = (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑘)(+_g‘(𝑅‘𝑘))(𝑔‘𝑘))))
36	24, 26, 35	3eqtr4g 2804	. . . 4 ⊢ (𝜑 → (+_𝑓‘𝑌) = (𝑧 ∈ ((Base‘𝑌) × (Base‘𝑌)) ↦ (𝑘 ∈ 𝐼 ↦ (((1^st ‘𝑧)‘𝑘)(+_g‘(𝑅‘𝑘))((2^nd ‘𝑧)‘𝑘)))))
37		eqid 2738	. . . . 5 ⊢ (∏_t‘(TopOpen ∘ 𝑅)) = (∏_t‘(TopOpen ∘ 𝑅))
38		eqid 2738	. . . . . . . 8 ⊢ (TopOpen‘𝑌) = (TopOpen‘𝑌)
39	15, 38	istps 21991	. . . . . . 7 ⊢ (𝑌 ∈ TopSp ↔ (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
40	14, 39	sylib 217	. . . . . 6 ⊢ (𝜑 → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
41		txtopon 22650	. . . . . 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 583	. . . . 5 ⊢ (𝜑 → ((TopOpen‘𝑌) ×_t (TopOpen‘𝑌)) ∈ (TopOn‘((Base‘𝑌) × (Base‘𝑌))))
43		topnfn 17053	. . . . . . . 8 ⊢ TopOpen Fn V
44		ssv 3941	. . . . . . . 8 ⊢ TopSp ⊆ V
45		fnssres 6539	. . . . . . . 8 ⊢ ((TopOpen Fn V ∧ TopSp ⊆ V) → (TopOpen ↾ TopSp) Fn TopSp)
46	43, 44, 45	mp2an 688	. . . . . . 7 ⊢ (TopOpen ↾ TopSp) Fn TopSp
47		fvres 6775	. . . . . . . . 9 ⊢ (𝑥 ∈ TopSp → ((TopOpen ↾ TopSp)‘𝑥) = (TopOpen‘𝑥))
48		eqid 2738	. . . . . . . . . 10 ⊢ (TopOpen‘𝑥) = (TopOpen‘𝑥)
49	48	tpstop 21994	. . . . . . . . 9 ⊢ (𝑥 ∈ TopSp → (TopOpen‘𝑥) ∈ Top)
50	47, 49	eqeltrd 2839	. . . . . . . 8 ⊢ (𝑥 ∈ TopSp → ((TopOpen ↾ TopSp)‘𝑥) ∈ Top)
51	50	rgen 3073	. . . . . . 7 ⊢ ∀𝑥 ∈ TopSp ((TopOpen ↾ TopSp)‘𝑥) ∈ Top
52		ffnfv 6974	. . . . . . 7 ⊢ ((TopOpen ↾ TopSp):TopSp⟶Top ↔ ((TopOpen ↾ TopSp) Fn TopSp ∧ ∀𝑥 ∈ TopSp ((TopOpen ↾ TopSp)‘𝑥) ∈ Top))
53	46, 51, 52	mpbir2an 707	. . . . . 6 ⊢ (TopOpen ↾ TopSp):TopSp⟶Top
54		fco2 6611	. . . . . 6 ⊢ (((TopOpen ↾ TopSp):TopSp⟶Top ∧ 𝑅:𝐼⟶TopSp) → (TopOpen ∘ 𝑅):𝐼⟶Top)
55	53, 13, 54	sylancr 586	. . . . 5 ⊢ (𝜑 → (TopOpen ∘ 𝑅):𝐼⟶Top)
56	33	mpompt 7366	. . . . . 6 ⊢ (𝑧 ∈ ((Base‘𝑌) × (Base‘𝑌)) ↦ (((1^st ‘𝑧)‘𝑘)(+_g‘(𝑅‘𝑘))((2^nd ‘𝑧)‘𝑘))) = (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ ((𝑓‘𝑘)(+_g‘(𝑅‘𝑘))(𝑔‘𝑘)))
57		eqid 2738	. . . . . . . 8 ⊢ (TopOpen‘(𝑅‘𝑘)) = (TopOpen‘(𝑅‘𝑘))
58		eqid 2738	. . . . . . . 8 ⊢ (+_g‘(𝑅‘𝑘)) = (+_g‘(𝑅‘𝑘))
59	4	ffvelrnda 6943	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑅‘𝑘) ∈ TopMnd)
60	40	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
61	60, 60	cnmpt1st 22727	. . . . . . . . 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 22687	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (TopOpen‘𝑌) = (∏_t‘(TopOpen ∘ 𝑅)))
63	62	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (TopOpen‘𝑌) = (∏_t‘(TopOpen ∘ 𝑅)))
64	63, 60	eqeltrrd 2840	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (∏_t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)))
65		toponuni 21971	. . . . . . . . . . . . 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 5165	. . . . . . . . . . 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 2738	. . . . . . . . . . . . 13 ⊢ ∪ (∏_t‘(TopOpen ∘ 𝑅)) = ∪ (∏_t‘(TopOpen ∘ 𝑅))
72	71, 37	ptpjcn 22670	. . . . . . . . . . . 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 1369	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ ∪ (∏_t‘(TopOpen ∘ 𝑅)) ↦ (𝑥‘𝑘)) ∈ ((∏_t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
74	67, 73	eqeltrd 2839	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥‘𝑘)) ∈ ((∏_t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
75	63	eqcomd 2744	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (∏_t‘(TopOpen ∘ 𝑅)) = (TopOpen‘𝑌))
76		fvco3 6849	. . . . . . . . . . . 12 ⊢ ((𝑅:𝐼⟶TopMnd ∧ 𝑘 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = (TopOpen‘(𝑅‘𝑘)))
77	4, 76	sylan 579	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = (TopOpen‘(𝑅‘𝑘)))
78	75, 77	oveq12d 7273	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((∏_t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)) = ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅‘𝑘))))
79	74, 78	eleqtrd 2841	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥‘𝑘)) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅‘𝑘))))
80		fveq1 6755	. . . . . . . . 9 ⊢ (𝑥 = 𝑓 → (𝑥‘𝑘) = (𝑓‘𝑘))
81	60, 60, 61, 60, 79, 80	cnmpt21 22730	. . . . . . . 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 22728	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ 𝑔) ∈ (((TopOpen‘𝑌) ×_t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)))
83		fveq1 6755	. . . . . . . . 9 ⊢ (𝑥 = 𝑔 → (𝑥‘𝑘) = (𝑔‘𝑘))
84	60, 60, 82, 60, 79, 83	cnmpt21 22730	. . . . . . . 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 23147	. . . . . . 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 7271	. . . . . . 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 2842	. . . . . 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 2843	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑧 ∈ ((Base‘𝑌) × (Base‘𝑌)) ↦ (((1^st ‘𝑧)‘𝑘)(+_g‘(𝑅‘𝑘))((2^nd ‘𝑧)‘𝑘))) ∈ (((TopOpen‘𝑌) ×_t (TopOpen‘𝑌)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
89	37, 42, 2, 55, 88	ptcn 22686	. . . 4 ⊢ (𝜑 → (𝑧 ∈ ((Base‘𝑌) × (Base‘𝑌)) ↦ (𝑘 ∈ 𝐼 ↦ (((1^st ‘𝑧)‘𝑘)(+_g‘(𝑅‘𝑘))((2^nd ‘𝑧)‘𝑘)))) ∈ (((TopOpen‘𝑌) ×_t (TopOpen‘𝑌)) Cn (∏_t‘(TopOpen ∘ 𝑅))))
90	36, 89	eqeltrd 2839	. . 3 ⊢ (𝜑 → (+_𝑓‘𝑌) ∈ (((TopOpen‘𝑌) ×_t (TopOpen‘𝑌)) Cn (∏_t‘(TopOpen ∘ 𝑅))))
91	62	oveq2d 7271	. . 3 ⊢ (𝜑 → (((TopOpen‘𝑌) ×_t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)) = (((TopOpen‘𝑌) ×_t (TopOpen‘𝑌)) Cn (∏_t‘(TopOpen ∘ 𝑅))))
92	90, 91	eleqtrrd 2842	. 2 ⊢ (𝜑 → (+_𝑓‘𝑌) ∈ (((TopOpen‘𝑌) ×_t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)))
93	25, 38	istmd 23133	. 2 ⊢ (𝑌 ∈ TopMnd ↔ (𝑌 ∈ Mnd ∧ 𝑌 ∈ TopSp ∧ (+_𝑓‘𝑌) ∈ (((TopOpen‘𝑌) ×_t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌))))
94	9, 14, 92, 93	syl3anbrc 1341	1 ⊢ (𝜑 → 𝑌 ∈ TopMnd)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ∀wral 3063 Vcvv 3422 ⊆ wss 3883 ⟨cop 4564 ∪ cuni 4836 ↦ cmpt 5153 × cxp 5578 ↾ cres 5582 ∘ ccom 5584 Fn wfn 6413 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ∈ cmpo 7257 1^st c1st 7802 2^nd c2nd 7803 Basecbs 16840 +_gcplusg 16888 TopOpenctopn 17049 ∏_tcpt 17066 X_scprds 17073 +_𝑓cplusf 18238 Mndcmnd 18300 Topctop 21950 TopOnctopon 21967 TopSpctps 21989 Cn ccn 22283 ×_t ctx 22619 TopMndctmd 23129

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fi 9100 df-sup 9131 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-fz 13169 df-struct 16776 df-slot 16811 df-ndx 16823 df-base 16841 df-plusg 16901 df-mulr 16902 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-topgen 17071 df-pt 17072 df-prds 17075 df-plusf 18240 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cn 22286 df-cnp 22287 df-tx 22621 df-tmd 23131

This theorem is referenced by: prdstgpd 23184

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