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Theorem prdstmdd 23512

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 23463	. . . . 5 ⊢ (𝑥 ∈ TopMnd → 𝑥 ∈ Mnd)
6	5	ssriv 3951	. . . 4 ⊢ TopMnd ⊆ Mnd
7		fss 6690	. . . 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 18603	. 2 ⊢ (𝜑 → 𝑌 ∈ Mnd)
10		tmdtps 23464	. . . . 5 ⊢ (𝑥 ∈ TopMnd → 𝑥 ∈ TopSp)
11	10	ssriv 3951	. . . 4 ⊢ TopMnd ⊆ TopSp
12		fss 6690	. . . 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 23017	. 2 ⊢ (𝜑 → 𝑌 ∈ TopSp)
15		eqid 2731	. . . . . . 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 6674	. . . . . . . 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 2731	. . . . . . 7 ⊢ (+_g‘𝑌) = (+_g‘𝑌)
23	1, 15, 16, 17, 19, 20, 21, 22	prdsplusgval 17369	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → (𝑓(+_g‘𝑌)𝑔) = (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑘)(+_g‘(𝑅‘𝑘))(𝑔‘𝑘))))
24	23	mpoeq3dva 7439	. . . . 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 2731	. . . . . 6 ⊢ (+_𝑓‘𝑌) = (+_𝑓‘𝑌)
26	15, 22, 25	plusffval 18517	. . . . 5 ⊢ (+_𝑓‘𝑌) = (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑓(+_g‘𝑌)𝑔))
27		vex 3450	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
28		vex 3450	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
29	27, 28	op1std 7936	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑓, 𝑔⟩ → (1^st ‘𝑧) = 𝑓)
30	29	fveq1d 6849	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑓, 𝑔⟩ → ((1^st ‘𝑧)‘𝑘) = (𝑓‘𝑘))
31	27, 28	op2ndd 7937	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑓, 𝑔⟩ → (2^nd ‘𝑧) = 𝑔)
32	31	fveq1d 6849	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑓, 𝑔⟩ → ((2^nd ‘𝑧)‘𝑘) = (𝑔‘𝑘))
33	30, 32	oveq12d 7380	. . . . . . 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 5212	. . . . . 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 7475	. . . . 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 2796	. . . 4 ⊢ (𝜑 → (+_𝑓‘𝑌) = (𝑧 ∈ ((Base‘𝑌) × (Base‘𝑌)) ↦ (𝑘 ∈ 𝐼 ↦ (((1^st ‘𝑧)‘𝑘)(+_g‘(𝑅‘𝑘))((2^nd ‘𝑧)‘𝑘)))))
37		eqid 2731	. . . . 5 ⊢ (∏_t‘(TopOpen ∘ 𝑅)) = (∏_t‘(TopOpen ∘ 𝑅))
38		eqid 2731	. . . . . . . 8 ⊢ (TopOpen‘𝑌) = (TopOpen‘𝑌)
39	15, 38	istps 22320	. . . . . . 7 ⊢ (𝑌 ∈ TopSp ↔ (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
40	14, 39	sylib 217	. . . . . 6 ⊢ (𝜑 → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
41		txtopon 22979	. . . . . 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 17321	. . . . . . . 8 ⊢ TopOpen Fn V
44		ssv 3971	. . . . . . . 8 ⊢ TopSp ⊆ V
45		fnssres 6629	. . . . . . . 8 ⊢ ((TopOpen Fn V ∧ TopSp ⊆ V) → (TopOpen ↾ TopSp) Fn TopSp)
46	43, 44, 45	mp2an 690	. . . . . . 7 ⊢ (TopOpen ↾ TopSp) Fn TopSp
47		fvres 6866	. . . . . . . . 9 ⊢ (𝑥 ∈ TopSp → ((TopOpen ↾ TopSp)‘𝑥) = (TopOpen‘𝑥))
48		eqid 2731	. . . . . . . . . 10 ⊢ (TopOpen‘𝑥) = (TopOpen‘𝑥)
49	48	tpstop 22323	. . . . . . . . 9 ⊢ (𝑥 ∈ TopSp → (TopOpen‘𝑥) ∈ Top)
50	47, 49	eqeltrd 2832	. . . . . . . 8 ⊢ (𝑥 ∈ TopSp → ((TopOpen ↾ TopSp)‘𝑥) ∈ Top)
51	50	rgen 3062	. . . . . . 7 ⊢ ∀𝑥 ∈ TopSp ((TopOpen ↾ TopSp)‘𝑥) ∈ Top
52		ffnfv 7071	. . . . . . 7 ⊢ ((TopOpen ↾ TopSp):TopSp⟶Top ↔ ((TopOpen ↾ TopSp) Fn TopSp ∧ ∀𝑥 ∈ TopSp ((TopOpen ↾ TopSp)‘𝑥) ∈ Top))
53	46, 51, 52	mpbir2an 709	. . . . . 6 ⊢ (TopOpen ↾ TopSp):TopSp⟶Top
54		fco2 6700	. . . . . 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 7475	. . . . . 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 2731	. . . . . . . 8 ⊢ (TopOpen‘(𝑅‘𝑘)) = (TopOpen‘(𝑅‘𝑘))
58		eqid 2731	. . . . . . . 8 ⊢ (+_g‘(𝑅‘𝑘)) = (+_g‘(𝑅‘𝑘))
59	4	ffvelcdmda 7040	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑅‘𝑘) ∈ TopMnd)
60	40	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
61	60, 60	cnmpt1st 23056	. . . . . . . . 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 23016	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (TopOpen‘𝑌) = (∏_t‘(TopOpen ∘ 𝑅)))
63	62	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (TopOpen‘𝑌) = (∏_t‘(TopOpen ∘ 𝑅)))
64	63, 60	eqeltrrd 2833	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (∏_t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)))
65		toponuni 22300	. . . . . . . . . . . . 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 5205	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥‘𝑘)) = (𝑥 ∈ ∪ (∏_t‘(TopOpen ∘ 𝑅)) ↦ (𝑥‘𝑘)))
68	2	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐼 ∈ 𝑊)
69	55	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (TopOpen ∘ 𝑅):𝐼⟶Top)
70		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝑘 ∈ 𝐼)
71		eqid 2731	. . . . . . . . . . . . 13 ⊢ ∪ (∏_t‘(TopOpen ∘ 𝑅)) = ∪ (∏_t‘(TopOpen ∘ 𝑅))
72	71, 37	ptpjcn 22999	. . . . . . . . . . . 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 1371	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ ∪ (∏_t‘(TopOpen ∘ 𝑅)) ↦ (𝑥‘𝑘)) ∈ ((∏_t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
74	67, 73	eqeltrd 2832	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥‘𝑘)) ∈ ((∏_t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
75	63	eqcomd 2737	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (∏_t‘(TopOpen ∘ 𝑅)) = (TopOpen‘𝑌))
76		fvco3 6945	. . . . . . . . . . . 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 7380	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((∏_t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)) = ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅‘𝑘))))
79	74, 78	eleqtrd 2834	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥‘𝑘)) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅‘𝑘))))
80		fveq1 6846	. . . . . . . . 9 ⊢ (𝑥 = 𝑓 → (𝑥‘𝑘) = (𝑓‘𝑘))
81	60, 60, 61, 60, 79, 80	cnmpt21 23059	. . . . . . . 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 23057	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ 𝑔) ∈ (((TopOpen‘𝑌) ×_t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)))
83		fveq1 6846	. . . . . . . . 9 ⊢ (𝑥 = 𝑔 → (𝑥‘𝑘) = (𝑔‘𝑘))
84	60, 60, 82, 60, 79, 83	cnmpt21 23059	. . . . . . . 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 23476	. . . . . . 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 7378	. . . . . . 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 2835	. . . . . 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 2836	. . . . 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 23015	. . . 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 2832	. . 3 ⊢ (𝜑 → (+_𝑓‘𝑌) ∈ (((TopOpen‘𝑌) ×_t (TopOpen‘𝑌)) Cn (∏_t‘(TopOpen ∘ 𝑅))))
91	62	oveq2d 7378	. . 3 ⊢ (𝜑 → (((TopOpen‘𝑌) ×_t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)) = (((TopOpen‘𝑌) ×_t (TopOpen‘𝑌)) Cn (∏_t‘(TopOpen ∘ 𝑅))))
92	90, 91	eleqtrrd 2835	. 2 ⊢ (𝜑 → (+_𝑓‘𝑌) ∈ (((TopOpen‘𝑌) ×_t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)))
93	25, 38	istmd 23462	. 2 ⊢ (𝑌 ∈ TopMnd ↔ (𝑌 ∈ Mnd ∧ 𝑌 ∈ TopSp ∧ (+_𝑓‘𝑌) ∈ (((TopOpen‘𝑌) ×_t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌))))
94	9, 14, 92, 93	syl3anbrc 1343	1 ⊢ (𝜑 → 𝑌 ∈ TopMnd)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3060 Vcvv 3446 ⊆ wss 3913 ⟨cop 4597 ∪ cuni 4870 ↦ cmpt 5193 × cxp 5636 ↾ cres 5640 ∘ ccom 5642 Fn wfn 6496 ⟶wf 6497 ‘cfv 6501 (class class class)co 7362 ∈ cmpo 7364 1^st c1st 7924 2^nd c2nd 7925 Basecbs 17094 +_gcplusg 17147 TopOpenctopn 17317 ∏_tcpt 17334 X_scprds 17341 +_𝑓cplusf 18508 Mndcmnd 18570 Topctop 22279 TopOnctopon 22296 TopSpctps 22318 Cn ccn 22612 ×_t ctx 22948 TopMndctmd 23458

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11116 ax-resscn 11117 ax-1cn 11118 ax-icn 11119 ax-addcl 11120 ax-addrcl 11121 ax-mulcl 11122 ax-mulrcl 11123 ax-mulcom 11124 ax-addass 11125 ax-mulass 11126 ax-distr 11127 ax-i2m1 11128 ax-1ne0 11129 ax-1rid 11130 ax-rnegex 11131 ax-rrecex 11132 ax-cnre 11133 ax-pre-lttri 11134 ax-pre-lttrn 11135 ax-pre-ltadd 11136 ax-pre-mulgt0 11137

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3448 df-sbc 3743 df-csb 3859 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-iin 4962 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-map 8774 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fi 9356 df-sup 9387 df-pnf 11200 df-mnf 11201 df-xr 11202 df-ltxr 11203 df-le 11204 df-sub 11396 df-neg 11397 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12423 df-z 12509 df-dec 12628 df-uz 12773 df-fz 13435 df-struct 17030 df-slot 17065 df-ndx 17077 df-base 17095 df-plusg 17160 df-mulr 17161 df-sca 17163 df-vsca 17164 df-ip 17165 df-tset 17166 df-ple 17167 df-ds 17169 df-hom 17171 df-cco 17172 df-rest 17318 df-topn 17319 df-0g 17337 df-topgen 17339 df-pt 17340 df-prds 17343 df-plusf 18510 df-mgm 18511 df-sgrp 18560 df-mnd 18571 df-top 22280 df-topon 22297 df-topsp 22319 df-bases 22333 df-cn 22615 df-cnp 22616 df-tx 22950 df-tmd 23460

This theorem is referenced by: prdstgpd 23513

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