prdstgpd - Metamath Proof Explorer

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Theorem prdstgpd 23636

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 = (𝑆X_s𝑅)
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 = (𝑆X_s𝑅)
2		prdstgpd.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
3		prdstgpd.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
4		prdstgpd.r	. . . 4 ⊢ (𝜑 → 𝑅:𝐼⟶TopGrp)
5		tgpgrp 23589	. . . . 5 ⊢ (𝑥 ∈ TopGrp → 𝑥 ∈ Grp)
6	5	ssriv 3986	. . . 4 ⊢ TopGrp ⊆ Grp
7		fss 6734	. . . 4 ⊢ ((𝑅:𝐼⟶TopGrp ∧ TopGrp ⊆ Grp) → 𝑅:𝐼⟶Grp)
8	4, 6, 7	sylancl 586	. . 3 ⊢ (𝜑 → 𝑅:𝐼⟶Grp)
9	1, 2, 3, 8	prdsgrpd 18935	. 2 ⊢ (𝜑 → 𝑌 ∈ Grp)
10		tgptmd 23590	. . . . 5 ⊢ (𝑥 ∈ TopGrp → 𝑥 ∈ TopMnd)
11	10	ssriv 3986	. . . 4 ⊢ TopGrp ⊆ TopMnd
12		fss 6734	. . . 4 ⊢ ((𝑅:𝐼⟶TopGrp ∧ TopGrp ⊆ TopMnd) → 𝑅:𝐼⟶TopMnd)
13	4, 11, 12	sylancl 586	. . 3 ⊢ (𝜑 → 𝑅:𝐼⟶TopMnd)
14	1, 2, 3, 13	prdstmdd 23635	. 2 ⊢ (𝜑 → 𝑌 ∈ TopMnd)
15		eqid 2732	. . . 4 ⊢ (∏_t‘(TopOpen ∘ 𝑅)) = (∏_t‘(TopOpen ∘ 𝑅))
16		eqid 2732	. . . . . 6 ⊢ (TopOpen‘𝑌) = (TopOpen‘𝑌)
17		eqid 2732	. . . . . 6 ⊢ (Base‘𝑌) = (Base‘𝑌)
18	16, 17	tmdtopon 23592	. . . . 5 ⊢ (𝑌 ∈ TopMnd → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
19	14, 18	syl 17	. . . 4 ⊢ (𝜑 → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
20		topnfn 17373	. . . . . 6 ⊢ TopOpen Fn V
21	4	ffnd 6718	. . . . . . 7 ⊢ (𝜑 → 𝑅 Fn 𝐼)
22		dffn2 6719	. . . . . . 7 ⊢ (𝑅 Fn 𝐼 ↔ 𝑅:𝐼⟶V)
23	21, 22	sylib 217	. . . . . 6 ⊢ (𝜑 → 𝑅:𝐼⟶V)
24		fnfco 6756	. . . . . 6 ⊢ ((TopOpen Fn V ∧ 𝑅:𝐼⟶V) → (TopOpen ∘ 𝑅) Fn 𝐼)
25	20, 23, 24	sylancr 587	. . . . 5 ⊢ (𝜑 → (TopOpen ∘ 𝑅) Fn 𝐼)
26		fvco3 6990	. . . . . . . 8 ⊢ ((𝑅:𝐼⟶TopGrp ∧ 𝑦 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅‘𝑦)))
27	4, 26	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅‘𝑦)))
28	4	ffvelcdmda 7086	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑅‘𝑦) ∈ TopGrp)
29		eqid 2732	. . . . . . . . 9 ⊢ (TopOpen‘(𝑅‘𝑦)) = (TopOpen‘(𝑅‘𝑦))
30		eqid 2732	. . . . . . . . 9 ⊢ (Base‘(𝑅‘𝑦)) = (Base‘(𝑅‘𝑦))
31	29, 30	tgptopon 23593	. . . . . . . 8 ⊢ ((𝑅‘𝑦) ∈ TopGrp → (TopOpen‘(𝑅‘𝑦)) ∈ (TopOn‘(Base‘(𝑅‘𝑦))))
32		topontop 22422	. . . . . . . 8 ⊢ ((TopOpen‘(𝑅‘𝑦)) ∈ (TopOn‘(Base‘(𝑅‘𝑦))) → (TopOpen‘(𝑅‘𝑦)) ∈ Top)
33	28, 31, 32	3syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (TopOpen‘(𝑅‘𝑦)) ∈ Top)
34	27, 33	eqeltrd 2833	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top)
35	34	ralrimiva 3146	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝐼 ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top)
36		ffnfv 7120	. . . . 5 ⊢ ((TopOpen ∘ 𝑅):𝐼⟶Top ↔ ((TopOpen ∘ 𝑅) Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top))
37	25, 35, 36	sylanbrc 583	. . . 4 ⊢ (𝜑 → (TopOpen ∘ 𝑅):𝐼⟶Top)
38	19	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
39	1, 3, 2, 21, 16	prdstopn 23139	. . . . . . . . . . . 12 ⊢ (𝜑 → (TopOpen‘𝑌) = (∏_t‘(TopOpen ∘ 𝑅)))
40	39	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (TopOpen‘𝑌) = (∏_t‘(TopOpen ∘ 𝑅)))
41	40	eqcomd 2738	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (∏_t‘(TopOpen ∘ 𝑅)) = (TopOpen‘𝑌))
42	41, 38	eqeltrd 2833	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (∏_t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)))
43		toponuni 22423	. . . . . . . . 9 ⊢ ((∏_t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)) → (Base‘𝑌) = ∪ (∏_t‘(TopOpen ∘ 𝑅)))
44		mpteq1 5241	. . . . . . . . 9 ⊢ ((Base‘𝑌) = ∪ (∏_t‘(TopOpen ∘ 𝑅)) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥‘𝑦)) = (𝑥 ∈ ∪ (∏_t‘(TopOpen ∘ 𝑅)) ↦ (𝑥‘𝑦)))
45	42, 43, 44	3syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥‘𝑦)) = (𝑥 ∈ ∪ (∏_t‘(TopOpen ∘ 𝑅)) ↦ (𝑥‘𝑦)))
46	2	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → 𝐼 ∈ 𝑊)
47	37	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (TopOpen ∘ 𝑅):𝐼⟶Top)
48		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → 𝑦 ∈ 𝐼)
49		eqid 2732	. . . . . . . . . 10 ⊢ ∪ (∏_t‘(TopOpen ∘ 𝑅)) = ∪ (∏_t‘(TopOpen ∘ 𝑅))
50	49, 15	ptpjcn 23122	. . . . . . . . 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 1371	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∈ ∪ (∏_t‘(TopOpen ∘ 𝑅)) ↦ (𝑥‘𝑦)) ∈ ((∏_t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
52	45, 51	eqeltrd 2833	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥‘𝑦)) ∈ ((∏_t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
53	41, 27	oveq12d 7429	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((∏_t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)) = ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅‘𝑦))))
54	52, 53	eleqtrd 2835	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥‘𝑦)) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅‘𝑦))))
55		eqid 2732	. . . . . . . 8 ⊢ (inv_g‘(𝑅‘𝑦)) = (inv_g‘(𝑅‘𝑦))
56	29, 55	tgpinv 23596	. . . . . . 7 ⊢ ((𝑅‘𝑦) ∈ TopGrp → (inv_g‘(𝑅‘𝑦)) ∈ ((TopOpen‘(𝑅‘𝑦)) Cn (TopOpen‘(𝑅‘𝑦))))
57	28, 56	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (inv_g‘(𝑅‘𝑦)) ∈ ((TopOpen‘(𝑅‘𝑦)) Cn (TopOpen‘(𝑅‘𝑦))))
58	38, 54, 57	cnmpt11f 23175	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ ((inv_g‘(𝑅‘𝑦))‘(𝑥‘𝑦))) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅‘𝑦))))
59	27	oveq2d 7427	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((TopOpen‘𝑌) Cn ((TopOpen ∘ 𝑅)‘𝑦)) = ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅‘𝑦))))
60	58, 59	eleqtrrd 2836	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ ((inv_g‘(𝑅‘𝑦))‘(𝑥‘𝑦))) ∈ ((TopOpen‘𝑌) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
61	15, 19, 2, 37, 60	ptcn 23138	. . 3 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑌) ↦ (𝑦 ∈ 𝐼 ↦ ((inv_g‘(𝑅‘𝑦))‘(𝑥‘𝑦)))) ∈ ((TopOpen‘𝑌) Cn (∏_t‘(TopOpen ∘ 𝑅))))
62		eqid 2732	. . . . . . 7 ⊢ (inv_g‘𝑌) = (inv_g‘𝑌)
63	17, 62	grpinvf 18873	. . . . . 6 ⊢ (𝑌 ∈ Grp → (inv_g‘𝑌):(Base‘𝑌)⟶(Base‘𝑌))
64	9, 63	syl 17	. . . . 5 ⊢ (𝜑 → (inv_g‘𝑌):(Base‘𝑌)⟶(Base‘𝑌))
65	64	feqmptd 6960	. . . 4 ⊢ (𝜑 → (inv_g‘𝑌) = (𝑥 ∈ (Base‘𝑌) ↦ ((inv_g‘𝑌)‘𝑥)))
66	2	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑌)) → 𝐼 ∈ 𝑊)
67	3	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑌)) → 𝑆 ∈ 𝑉)
68	8	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑌)) → 𝑅:𝐼⟶Grp)
69		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑌)) → 𝑥 ∈ (Base‘𝑌))
70	1, 66, 67, 68, 17, 62, 69	prdsinvgd 18936	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑌)) → ((inv_g‘𝑌)‘𝑥) = (𝑦 ∈ 𝐼 ↦ ((inv_g‘(𝑅‘𝑦))‘(𝑥‘𝑦))))
71	70	mpteq2dva 5248	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑌) ↦ ((inv_g‘𝑌)‘𝑥)) = (𝑥 ∈ (Base‘𝑌) ↦ (𝑦 ∈ 𝐼 ↦ ((inv_g‘(𝑅‘𝑦))‘(𝑥‘𝑦)))))
72	65, 71	eqtrd 2772	. . 3 ⊢ (𝜑 → (inv_g‘𝑌) = (𝑥 ∈ (Base‘𝑌) ↦ (𝑦 ∈ 𝐼 ↦ ((inv_g‘(𝑅‘𝑦))‘(𝑥‘𝑦)))))
73	39	oveq2d 7427	. . 3 ⊢ (𝜑 → ((TopOpen‘𝑌) Cn (TopOpen‘𝑌)) = ((TopOpen‘𝑌) Cn (∏_t‘(TopOpen ∘ 𝑅))))
74	61, 72, 73	3eltr4d 2848	. 2 ⊢ (𝜑 → (inv_g‘𝑌) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘𝑌)))
75	16, 62	istgp 23588	. 2 ⊢ (𝑌 ∈ TopGrp ↔ (𝑌 ∈ Grp ∧ 𝑌 ∈ TopMnd ∧ (inv_g‘𝑌) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘𝑌))))
76	9, 14, 74, 75	syl3anbrc 1343	1 ⊢ (𝜑 → 𝑌 ∈ TopGrp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 Vcvv 3474 ⊆ wss 3948 ∪ cuni 4908 ↦ cmpt 5231 ∘ ccom 5680 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 Basecbs 17146 TopOpenctopn 17369 ∏_tcpt 17386 X_scprds 17393 Grpcgrp 18821 inv_gcminusg 18822 Topctop 22402 TopOnctopon 22419 Cn ccn 22735 TopMndctmd 23581 TopGrpctgp 23582

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 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 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 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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 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 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-z 12561 df-dec 12680 df-uz 12825 df-fz 13487 df-struct 17082 df-slot 17117 df-ndx 17129 df-base 17147 df-plusg 17212 df-mulr 17213 df-sca 17215 df-vsca 17216 df-ip 17217 df-tset 17218 df-ple 17219 df-ds 17221 df-hom 17223 df-cco 17224 df-rest 17370 df-topn 17371 df-0g 17389 df-topgen 17391 df-pt 17392 df-prds 17395 df-plusf 18562 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-grp 18824 df-minusg 18825 df-top 22403 df-topon 22420 df-topsp 22442 df-bases 22456 df-cn 22738 df-cnp 22739 df-tx 23073 df-tmd 23583 df-tgp 23584

This theorem is referenced by: (None)

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