pttopon - Metamath Proof Explorer

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

Theorem pttopon 23544

Description: The base set for the product topology. (Contributed by Mario Carneiro, 22-Aug-2015.)

**Hypothesis**
Ref	Expression
ptunimpt.j	⊢ 𝐽 = (∏_t‘(𝑥 ∈ 𝐴 ↦ 𝐾))

**Assertion**
Ref	Expression
pttopon	⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐾 ∈ (TopOn‘𝐵)) → 𝐽 ∈ (TopOn‘X𝑥 ∈ 𝐴 𝐵))

Distinct variable group: 𝑥,𝐴 Allowed substitution hints: 𝐵(𝑥) 𝐽(𝑥) 𝐾(𝑥) 𝑉(𝑥)

**Proof of Theorem pttopon**
Step	Hyp	Ref	Expression
1		topontop 22859	. . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝐵) → 𝐾 ∈ Top)
2	1	ralimi 3072	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐾 ∈ (TopOn‘𝐵) → ∀𝑥 ∈ 𝐴 𝐾 ∈ Top)
3		eqid 2725	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐾) = (𝑥 ∈ 𝐴 ↦ 𝐾)
4	3	fmpt 7119	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐾 ∈ Top ↔ (𝑥 ∈ 𝐴 ↦ 𝐾):𝐴⟶Top)
5	2, 4	sylib 217	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐾 ∈ (TopOn‘𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐾):𝐴⟶Top)
6		ptunimpt.j	. . . 4 ⊢ 𝐽 = (∏_t‘(𝑥 ∈ 𝐴 ↦ 𝐾))
7		pttop 23530	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐾):𝐴⟶Top) → (∏_t‘(𝑥 ∈ 𝐴 ↦ 𝐾)) ∈ Top)
8	6, 7	eqeltrid 2829	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐾):𝐴⟶Top) → 𝐽 ∈ Top)
9	5, 8	sylan2 591	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐾 ∈ (TopOn‘𝐵)) → 𝐽 ∈ Top)
10		toponuni 22860	. . . . . 6 ⊢ (𝐾 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐾)
11	10	ralimi 3072	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐾 ∈ (TopOn‘𝐵) → ∀𝑥 ∈ 𝐴 𝐵 = ∪ 𝐾)
12		ixpeq2 8930	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = ∪ 𝐾 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 ∪ 𝐾)
13	11, 12	syl 17	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐾 ∈ (TopOn‘𝐵) → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 ∪ 𝐾)
14	13	adantl 480	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐾 ∈ (TopOn‘𝐵)) → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 ∪ 𝐾)
15	6	ptunimpt 23543	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐾 ∈ Top) → X𝑥 ∈ 𝐴 ∪ 𝐾 = ∪ 𝐽)
16	2, 15	sylan2 591	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐾 ∈ (TopOn‘𝐵)) → X𝑥 ∈ 𝐴 ∪ 𝐾 = ∪ 𝐽)
17	14, 16	eqtrd 2765	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐾 ∈ (TopOn‘𝐵)) → X𝑥 ∈ 𝐴 𝐵 = ∪ 𝐽)
18		istopon 22858	. 2 ⊢ (𝐽 ∈ (TopOn‘X𝑥 ∈ 𝐴 𝐵) ↔ (𝐽 ∈ Top ∧ X𝑥 ∈ 𝐴 𝐵 = ∪ 𝐽))
19	9, 17, 18	sylanbrc 581	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐾 ∈ (TopOn‘𝐵)) → 𝐽 ∈ (TopOn‘X𝑥 ∈ 𝐴 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3050 ∪ cuni 4909 ↦ cmpt 5232 ⟶wf 6545 ‘cfv 6549 Xcixp 8916 ∏_tcpt 17423 Topctop 22839 TopOnctopon 22856

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-om 7872 df-1o 8487 df-er 8725 df-ixp 8917 df-en 8965 df-fin 8968 df-fi 9436 df-topgen 17428 df-pt 17429 df-top 22840 df-topon 22857 df-bases 22893

This theorem is referenced by: pttoponconst 23545 ptclsg 23563 dfac14lem 23565 ptcnp 23570 prdstps 23577

W3C validator