Theorem prdstgpd 24405

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

Hypotheses

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

Assertion

Ref	Expression
prdstgpd	⊢ (𝜑 → 𝑌 ∈ TopGrp)

Proof of Theorem prdstgpd

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prdstgpd.y	. . 3 ⊢ 𝑌 = (𝑆Xs𝑅)
2		prdstgpd.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
3		prdstgpd.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
4		prdstgpd.r	. . . 4 ⊢ (𝜑 → 𝑅:𝐼⟶TopGrp)
5		tgpgrp 24358	. . . . 5 ⊢ (𝑥 ∈ TopGrp → 𝑥 ∈ Grp)
6	5	ssriv 3934	. . . 4 ⊢ TopGrp ⊆ Grp
7		fss 6714	. . . 4 ⊢ ((𝑅:𝐼⟶TopGrp ∧ TopGrp ⊆ Grp) → 𝑅:𝐼⟶Grp)
8	4, 6, 7	sylancl 598	. . 3 ⊢ (𝜑 → 𝑅:𝐼⟶Grp)
9	1, 2, 3, 8	prdsgrpd 19221	. 2 ⊢ (𝜑 → 𝑌 ∈ Grp)
10		tgptmd 24359	. . . . 5 ⊢ (𝑥 ∈ TopGrp → 𝑥 ∈ TopMnd)
11	10	ssriv 3934	. . . 4 ⊢ TopGrp ⊆ TopMnd
12		fss 6714	. . . 4 ⊢ ((𝑅:𝐼⟶TopGrp ∧ TopGrp ⊆ TopMnd) → 𝑅:𝐼⟶TopMnd)
13	4, 11, 12	sylancl 598	. . 3 ⊢ (𝜑 → 𝑅:𝐼⟶TopMnd)
14	1, 2, 3, 13	prdstmdd 24404	. 2 ⊢ (𝜑 → 𝑌 ∈ TopMnd)
15		eqid 2760	. . . 4 ⊢ (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅))
16		eqid 2760	. . . . . 6 ⊢ (TopOpen‘𝑌) = (TopOpen‘𝑌)
17		eqid 2760	. . . . . 6 ⊢ (Base‘𝑌) = (Base‘𝑌)
18	16, 17	tmdtopon 24361	. . . . 5 ⊢ (𝑌 ∈ TopMnd → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
19	14, 18	syl 18	. . . 4 ⊢ (𝜑 → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
20		topnfn 17557	. . . . . 6 ⊢ TopOpen Fn V
21	4	ffnd 6698	. . . . . . 7 ⊢ (𝜑 → 𝑅 Fn 𝐼)
22		dffn2 6699	. . . . . . 7 ⊢ (𝑅 Fn 𝐼 ↔ 𝑅:𝐼⟶V)
23	21, 22	sylib 221	. . . . . 6 ⊢ (𝜑 → 𝑅:𝐼⟶V)
24		fnfco 6735	. . . . . 6 ⊢ ((TopOpen Fn V ∧ 𝑅:𝐼⟶V) → (TopOpen ∘ 𝑅) Fn 𝐼)
25	20, 23, 24	sylancr 599	. . . . 5 ⊢ (𝜑 → (TopOpen ∘ 𝑅) Fn 𝐼)
26		fvco3 6973	. . . . . . . 8 ⊢ ((𝑅:𝐼⟶TopGrp ∧ 𝑦 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅‘𝑦)))
27	4, 26	sylan 592	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅‘𝑦)))
28	4	ffvelcdmda 7072	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑅‘𝑦) ∈ TopGrp)
29		eqid 2760	. . . . . . . . 9 ⊢ (TopOpen‘(𝑅‘𝑦)) = (TopOpen‘(𝑅‘𝑦))
30		eqid 2760	. . . . . . . . 9 ⊢ (Base‘(𝑅‘𝑦)) = (Base‘(𝑅‘𝑦))
31	29, 30	tgptopon 24362	. . . . . . . 8 ⊢ ((𝑅‘𝑦) ∈ TopGrp → (TopOpen‘(𝑅‘𝑦)) ∈ (TopOn‘(Base‘(𝑅‘𝑦))))
32		topontop 23192	. . . . . . . 8 ⊢ ((TopOpen‘(𝑅‘𝑦)) ∈ (TopOn‘(Base‘(𝑅‘𝑦))) → (TopOpen‘(𝑅‘𝑦)) ∈ Top)
33	28, 31, 32	3syl 19	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (TopOpen‘(𝑅‘𝑦)) ∈ Top)
34	27, 33	eqeltrd 2860	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top)
35	34	ralrimiva 3154	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝐼 ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top)
36		ffnfv 7107	. . . . 5 ⊢ ((TopOpen ∘ 𝑅):𝐼⟶Top ↔ ((TopOpen ∘ 𝑅) Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top))
37	25, 35, 36	sylanbrc 595	. . . 4 ⊢ (𝜑 → (TopOpen ∘ 𝑅):𝐼⟶Top)
38	19	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
39	1, 3, 2, 21, 16	prdstopn 23908	. . . . . . . . . . . 12 ⊢ (𝜑 → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
40	39	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
41	40	eqcomd 2766	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (∏t‘(TopOpen ∘ 𝑅)) = (TopOpen‘𝑌))
42	41, 38	eqeltrd 2860	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)))
43		toponuni 23193	. . . . . . . . 9 ⊢ ((∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)) → (Base‘𝑌) = ∪ (∏t‘(TopOpen ∘ 𝑅)))
44		mpteq1 5193	. . . . . . . . 9 ⊢ ((Base‘𝑌) = ∪ (∏t‘(TopOpen ∘ 𝑅)) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥‘𝑦)) = (𝑥 ∈ ∪ (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥‘𝑦)))
45	42, 43, 44	3syl 19	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥‘𝑦)) = (𝑥 ∈ ∪ (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥‘𝑦)))
46	2	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → 𝐼 ∈ 𝑊)
47	37	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (TopOpen ∘ 𝑅):𝐼⟶Top)
48		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → 𝑦 ∈ 𝐼)
49		eqid 2760	. . . . . . . . . 10 ⊢ ∪ (∏t‘(TopOpen ∘ 𝑅)) = ∪ (∏t‘(TopOpen ∘ 𝑅))
50	49, 15	ptpjcn 23891	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑊 ∧ (TopOpen ∘ 𝑅):𝐼⟶Top ∧ 𝑦 ∈ 𝐼) → (𝑥 ∈ ∪ (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥‘𝑦)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
51	46, 47, 48, 50	syl3anc 1398	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∈ ∪ (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥‘𝑦)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
52	45, 51	eqeltrd 2860	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥‘𝑦)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
53	41, 27	oveq12d 7426	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)) = ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅‘𝑦))))
54	52, 53	eleqtrd 2862	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥‘𝑦)) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅‘𝑦))))
55		eqid 2760	. . . . . . . 8 ⊢ (invg‘(𝑅‘𝑦)) = (invg‘(𝑅‘𝑦))
56	29, 55	tgpinv 24365	. . . . . . 7 ⊢ ((𝑅‘𝑦) ∈ TopGrp → (invg‘(𝑅‘𝑦)) ∈ ((TopOpen‘(𝑅‘𝑦)) Cn (TopOpen‘(𝑅‘𝑦))))
57	28, 56	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (invg‘(𝑅‘𝑦)) ∈ ((TopOpen‘(𝑅‘𝑦)) Cn (TopOpen‘(𝑅‘𝑦))))
58	38, 54, 57	cnmpt11f 23944	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ ((invg‘(𝑅‘𝑦))‘(𝑥‘𝑦))) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅‘𝑦))))
59	27	oveq2d 7424	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((TopOpen‘𝑌) Cn ((TopOpen ∘ 𝑅)‘𝑦)) = ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅‘𝑦))))
60	58, 59	eleqtrrd 2863	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ ((invg‘(𝑅‘𝑦))‘(𝑥‘𝑦))) ∈ ((TopOpen‘𝑌) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
61	15, 19, 2, 37, 60	ptcn 23907	. . 3 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑌) ↦ (𝑦 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑦))‘(𝑥‘𝑦)))) ∈ ((TopOpen‘𝑌) Cn (∏t‘(TopOpen ∘ 𝑅))))
62		eqid 2760	. . . . . . 7 ⊢ (invg‘𝑌) = (invg‘𝑌)
63	17, 62	grpinvf 19158	. . . . . 6 ⊢ (𝑌 ∈ Grp → (invg‘𝑌):(Base‘𝑌)⟶(Base‘𝑌))
64	9, 63	syl 18	. . . . 5 ⊢ (𝜑 → (invg‘𝑌):(Base‘𝑌)⟶(Base‘𝑌))
65	64	feqmptd 6941	. . . 4 ⊢ (𝜑 → (invg‘𝑌) = (𝑥 ∈ (Base‘𝑌) ↦ ((invg‘𝑌)‘𝑥)))
66	2	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑌)) → 𝐼 ∈ 𝑊)
67	3	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑌)) → 𝑆 ∈ 𝑉)
68	8	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑌)) → 𝑅:𝐼⟶Grp)
69		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑌)) → 𝑥 ∈ (Base‘𝑌))
70	1, 66, 67, 68, 17, 62, 69	prdsinvgd 19222	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑌)) → ((invg‘𝑌)‘𝑥) = (𝑦 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑦))‘(𝑥‘𝑦))))
71	70	mpteq2dva 5197	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑌) ↦ ((invg‘𝑌)‘𝑥)) = (𝑥 ∈ (Base‘𝑌) ↦ (𝑦 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑦))‘(𝑥‘𝑦)))))
72	65, 71	eqtrd 2795	. . 3 ⊢ (𝜑 → (invg‘𝑌) = (𝑥 ∈ (Base‘𝑌) ↦ (𝑦 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑦))‘(𝑥‘𝑦)))))
73	39	oveq2d 7424	. . 3 ⊢ (𝜑 → ((TopOpen‘𝑌) Cn (TopOpen‘𝑌)) = ((TopOpen‘𝑌) Cn (∏t‘(TopOpen ∘ 𝑅))))
74	61, 72, 73	3eltr4d 2875	. 2 ⊢ (𝜑 → (invg‘𝑌) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘𝑌)))
75	16, 62	istgp 24357	. 2 ⊢ (𝑌 ∈ TopGrp ↔ (𝑌 ∈ Grp ∧ 𝑌 ∈ TopMnd ∧ (invg‘𝑌) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘𝑌))))
76	9, 14, 74, 75	syl3anbrc 1362	1 ⊢ (𝜑 → 𝑌 ∈ TopGrp)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  Vcvv 3450   ⊆ wss 3898  ∪ cuni 4866   ↦ cmpt 5185   ∘ ccom 5651   Fn wfn 6522  ⟶wf 6523  ‘cfv 6527  (class class class)co 7408  Basecbs 17348  TopOpenctopn 17553  ∏_tcpt 17570  X_scprds 17577  Grpcgrp 19105  inv_gcminusg 19106  Topctop 23172  TopOnctopon 23189   Cn ccn 23503  TopMndctmd 24350  TopGrpctgp 24351

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-cnex 11227  ax-resscn 11228  ax-1cn 11229  ax-icn 11230  ax-addcl 11231  ax-addrcl 11232  ax-mulcl 11233  ax-mulrcl 11234  ax-mulcom 11235  ax-addass 11236  ax-mulass 11237  ax-distr 11238  ax-i2m1 11239  ax-1ne0 11240  ax-1rid 11241  ax-rnegex 11242  ax-rrecex 11243  ax-cnre 11244  ax-pre-lttri 11245  ax-pre-lttrn 11246  ax-pre-ltadd 11247  ax-pre-mulgt0 11248

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-2o 8455  df-er 8695  df-map 8827  df-ixp 8904  df-en 8952  df-dom 8953  df-sdom 8954  df-fin 8955  df-fi 9381  df-sup 9412  df-pnf 11316  df-mnf 11317  df-xr 11318  df-ltxr 11319  df-le 11320  df-sub 11514  df-neg 11515  df-nn 12305  df-2 12374  df-3 12375  df-4 12376  df-5 12377  df-6 12378  df-7 12379  df-8 12380  df-9 12381  df-n0 12576  df-z 12663  df-dec 12784  df-uz 12935  df-fz 13609  df-struct 17286  df-slot 17321  df-ndx 17333  df-base 17349  df-plusg 17402  df-mulr 17403  df-sca 17405  df-vsca 17406  df-ip 17407  df-tset 17408  df-ple 17409  df-ds 17411  df-hom 17413  df-cco 17414  df-rest 17554  df-topn 17555  df-0g 17573  df-topgen 17575  df-pt 17576  df-prds 17579  df-plusf 18776  df-mgm 18777  df-sgrp 18869  df-mnd 18885  df-grp 19108  df-minusg 19109  df-top 23173  df-topon 23190  df-topsp 23212  df-bases 23225  df-cn 23506  df-cnp 23507  df-tx 23842  df-tmd 24352  df-tgp 24353

This theorem is used by: (None)