prdstmdd - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > prdstmdd		Structured version Visualization version GIF version

Theorem prdstmdd 23848

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 23799	. . . . 5 ⊢ (𝑥 ∈ TopMnd → 𝑥 ∈ Mnd)
6	5	ssriv 3985	. . . 4 ⊢ TopMnd ⊆ Mnd
7		fss 6733	. . . 4 ⊢ ((𝑅:𝐼⟶TopMnd ∧ TopMnd ⊆ Mnd) → 𝑅:𝐼⟶Mnd)
8	4, 6, 7	sylancl 584	. . 3 ⊢ (𝜑 → 𝑅:𝐼⟶Mnd)
9	1, 2, 3, 8	prdsmndd 18692	. 2 ⊢ (𝜑 → 𝑌 ∈ Mnd)
10		tmdtps 23800	. . . . 5 ⊢ (𝑥 ∈ TopMnd → 𝑥 ∈ TopSp)
11	10	ssriv 3985	. . . 4 ⊢ TopMnd ⊆ TopSp
12		fss 6733	. . . 4 ⊢ ((𝑅:𝐼⟶TopMnd ∧ TopMnd ⊆ TopSp) → 𝑅:𝐼⟶TopSp)
13	4, 11, 12	sylancl 584	. . 3 ⊢ (𝜑 → 𝑅:𝐼⟶TopSp)
14	1, 3, 2, 13	prdstps 23353	. 2 ⊢ (𝜑 → 𝑌 ∈ TopSp)
15		eqid 2730	. . . . . . 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 6717	. . . . . . . 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 2730	. . . . . . 7 ⊢ (+_g‘𝑌) = (+_g‘𝑌)
23	1, 15, 16, 17, 19, 20, 21, 22	prdsplusgval 17423	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → (𝑓(+_g‘𝑌)𝑔) = (𝑘 ∈ 𝐼 ↦ ((𝑓‘𝑘)(+_g‘(𝑅‘𝑘))(𝑔‘𝑘))))
24	23	mpoeq3dva 7488	. . . . 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 2730	. . . . . 6 ⊢ (+_𝑓‘𝑌) = (+_𝑓‘𝑌)
26	15, 22, 25	plusffval 18571	. . . . 5 ⊢ (+_𝑓‘𝑌) = (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑓(+_g‘𝑌)𝑔))
27		vex 3476	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
28		vex 3476	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
29	27, 28	op1std 7987	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑓, 𝑔⟩ → (1^st ‘𝑧) = 𝑓)
30	29	fveq1d 6892	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑓, 𝑔⟩ → ((1^st ‘𝑧)‘𝑘) = (𝑓‘𝑘))
31	27, 28	op2ndd 7988	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑓, 𝑔⟩ → (2^nd ‘𝑧) = 𝑔)
32	31	fveq1d 6892	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑓, 𝑔⟩ → ((2^nd ‘𝑧)‘𝑘) = (𝑔‘𝑘))
33	30, 32	oveq12d 7429	. . . . . . 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 5249	. . . . . 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 7524	. . . . 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 2795	. . . 4 ⊢ (𝜑 → (+_𝑓‘𝑌) = (𝑧 ∈ ((Base‘𝑌) × (Base‘𝑌)) ↦ (𝑘 ∈ 𝐼 ↦ (((1^st ‘𝑧)‘𝑘)(+_g‘(𝑅‘𝑘))((2^nd ‘𝑧)‘𝑘)))))
37		eqid 2730	. . . . 5 ⊢ (∏_t‘(TopOpen ∘ 𝑅)) = (∏_t‘(TopOpen ∘ 𝑅))
38		eqid 2730	. . . . . . . 8 ⊢ (TopOpen‘𝑌) = (TopOpen‘𝑌)
39	15, 38	istps 22656	. . . . . . 7 ⊢ (𝑌 ∈ TopSp ↔ (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
40	14, 39	sylib 217	. . . . . 6 ⊢ (𝜑 → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
41		txtopon 23315	. . . . . 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 582	. . . . 5 ⊢ (𝜑 → ((TopOpen‘𝑌) ×_t (TopOpen‘𝑌)) ∈ (TopOn‘((Base‘𝑌) × (Base‘𝑌))))
43		topnfn 17375	. . . . . . . 8 ⊢ TopOpen Fn V
44		ssv 4005	. . . . . . . 8 ⊢ TopSp ⊆ V
45		fnssres 6672	. . . . . . . 8 ⊢ ((TopOpen Fn V ∧ TopSp ⊆ V) → (TopOpen ↾ TopSp) Fn TopSp)
46	43, 44, 45	mp2an 688	. . . . . . 7 ⊢ (TopOpen ↾ TopSp) Fn TopSp
47		fvres 6909	. . . . . . . . 9 ⊢ (𝑥 ∈ TopSp → ((TopOpen ↾ TopSp)‘𝑥) = (TopOpen‘𝑥))
48		eqid 2730	. . . . . . . . . 10 ⊢ (TopOpen‘𝑥) = (TopOpen‘𝑥)
49	48	tpstop 22659	. . . . . . . . 9 ⊢ (𝑥 ∈ TopSp → (TopOpen‘𝑥) ∈ Top)
50	47, 49	eqeltrd 2831	. . . . . . . 8 ⊢ (𝑥 ∈ TopSp → ((TopOpen ↾ TopSp)‘𝑥) ∈ Top)
51	50	rgen 3061	. . . . . . 7 ⊢ ∀𝑥 ∈ TopSp ((TopOpen ↾ TopSp)‘𝑥) ∈ Top
52		ffnfv 7119	. . . . . . 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 6743	. . . . . 6 ⊢ (((TopOpen ↾ TopSp):TopSp⟶Top ∧ 𝑅:𝐼⟶TopSp) → (TopOpen ∘ 𝑅):𝐼⟶Top)
55	53, 13, 54	sylancr 585	. . . . 5 ⊢ (𝜑 → (TopOpen ∘ 𝑅):𝐼⟶Top)
56	33	mpompt 7524	. . . . . 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 2730	. . . . . . . 8 ⊢ (TopOpen‘(𝑅‘𝑘)) = (TopOpen‘(𝑅‘𝑘))
58		eqid 2730	. . . . . . . 8 ⊢ (+_g‘(𝑅‘𝑘)) = (+_g‘(𝑅‘𝑘))
59	4	ffvelcdmda 7085	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑅‘𝑘) ∈ TopMnd)
60	40	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
61	60, 60	cnmpt1st 23392	. . . . . . . . 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 23352	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (TopOpen‘𝑌) = (∏_t‘(TopOpen ∘ 𝑅)))
63	62	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (TopOpen‘𝑌) = (∏_t‘(TopOpen ∘ 𝑅)))
64	63, 60	eqeltrrd 2832	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (∏_t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)))
65		toponuni 22636	. . . . . . . . . . . . 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 5242	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥‘𝑘)) = (𝑥 ∈ ∪ (∏_t‘(TopOpen ∘ 𝑅)) ↦ (𝑥‘𝑘)))
68	2	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐼 ∈ 𝑊)
69	55	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (TopOpen ∘ 𝑅):𝐼⟶Top)
70		simpr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝑘 ∈ 𝐼)
71		eqid 2730	. . . . . . . . . . . . 13 ⊢ ∪ (∏_t‘(TopOpen ∘ 𝑅)) = ∪ (∏_t‘(TopOpen ∘ 𝑅))
72	71, 37	ptpjcn 23335	. . . . . . . . . . . 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 2831	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥‘𝑘)) ∈ ((∏_t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
75	63	eqcomd 2736	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (∏_t‘(TopOpen ∘ 𝑅)) = (TopOpen‘𝑌))
76		fvco3 6989	. . . . . . . . . . . 12 ⊢ ((𝑅:𝐼⟶TopMnd ∧ 𝑘 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = (TopOpen‘(𝑅‘𝑘)))
77	4, 76	sylan 578	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = (TopOpen‘(𝑅‘𝑘)))
78	75, 77	oveq12d 7429	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((∏_t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)) = ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅‘𝑘))))
79	74, 78	eleqtrd 2833	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥‘𝑘)) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅‘𝑘))))
80		fveq1 6889	. . . . . . . . 9 ⊢ (𝑥 = 𝑓 → (𝑥‘𝑘) = (𝑓‘𝑘))
81	60, 60, 61, 60, 79, 80	cnmpt21 23395	. . . . . . . 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 23393	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ 𝑔) ∈ (((TopOpen‘𝑌) ×_t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)))
83		fveq1 6889	. . . . . . . . 9 ⊢ (𝑥 = 𝑔 → (𝑥‘𝑘) = (𝑔‘𝑘))
84	60, 60, 82, 60, 79, 83	cnmpt21 23395	. . . . . . . 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 23812	. . . . . . 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 7427	. . . . . . 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 2834	. . . . . 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 2835	. . . . 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 23351	. . . 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 2831	. . 3 ⊢ (𝜑 → (+_𝑓‘𝑌) ∈ (((TopOpen‘𝑌) ×_t (TopOpen‘𝑌)) Cn (∏_t‘(TopOpen ∘ 𝑅))))
91	62	oveq2d 7427	. . 3 ⊢ (𝜑 → (((TopOpen‘𝑌) ×_t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)) = (((TopOpen‘𝑌) ×_t (TopOpen‘𝑌)) Cn (∏_t‘(TopOpen ∘ 𝑅))))
92	90, 91	eleqtrrd 2834	. 2 ⊢ (𝜑 → (+_𝑓‘𝑌) ∈ (((TopOpen‘𝑌) ×_t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)))
93	25, 38	istmd 23798	. 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 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ∀wral 3059 Vcvv 3472 ⊆ wss 3947 ⟨cop 4633 ∪ cuni 4907 ↦ cmpt 5230 × cxp 5673 ↾ cres 5677 ∘ ccom 5679 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 ∈ cmpo 7413 1^st c1st 7975 2^nd c2nd 7976 Basecbs 17148 +_gcplusg 17201 TopOpenctopn 17371 ∏_tcpt 17388 X_scprds 17395 +_𝑓cplusf 18562 Mndcmnd 18659 Topctop 22615 TopOnctopon 22632 TopSpctps 22654 Cn ccn 22948 ×_t ctx 23284 TopMndctmd 23794

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fi 9408 df-sup 9439 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13489 df-struct 17084 df-slot 17119 df-ndx 17131 df-base 17149 df-plusg 17214 df-mulr 17215 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-hom 17225 df-cco 17226 df-rest 17372 df-topn 17373 df-0g 17391 df-topgen 17393 df-pt 17394 df-prds 17397 df-plusf 18564 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-top 22616 df-topon 22633 df-topsp 22655 df-bases 22669 df-cn 22951 df-cnp 22952 df-tx 23286 df-tmd 23796

This theorem is referenced by: prdstgpd 23849

W3C validator