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Theorem pttopon 23532

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 22849	. . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝐵) → 𝐾 ∈ Top)
2	1	ralimi 3073	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐾 ∈ (TopOn‘𝐵) → ∀𝑥 ∈ 𝐴 𝐾 ∈ Top)
3		eqid 2735	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐾) = (𝑥 ∈ 𝐴 ↦ 𝐾)
4	3	fmpt 7099	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐾 ∈ Top ↔ (𝑥 ∈ 𝐴 ↦ 𝐾):𝐴⟶Top)
5	2, 4	sylib 218	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐾 ∈ (TopOn‘𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐾):𝐴⟶Top)
6		ptunimpt.j	. . . 4 ⊢ 𝐽 = (∏_t‘(𝑥 ∈ 𝐴 ↦ 𝐾))
7		pttop 23518	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐾):𝐴⟶Top) → (∏_t‘(𝑥 ∈ 𝐴 ↦ 𝐾)) ∈ Top)
8	6, 7	eqeltrid 2838	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐾):𝐴⟶Top) → 𝐽 ∈ Top)
9	5, 8	sylan2 593	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐾 ∈ (TopOn‘𝐵)) → 𝐽 ∈ Top)
10		toponuni 22850	. . . . . 6 ⊢ (𝐾 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐾)
11	10	ralimi 3073	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐾 ∈ (TopOn‘𝐵) → ∀𝑥 ∈ 𝐴 𝐵 = ∪ 𝐾)
12		ixpeq2 8923	. . . . 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 481	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐾 ∈ (TopOn‘𝐵)) → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 ∪ 𝐾)
15	6	ptunimpt 23531	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐾 ∈ Top) → X𝑥 ∈ 𝐴 ∪ 𝐾 = ∪ 𝐽)
16	2, 15	sylan2 593	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐾 ∈ (TopOn‘𝐵)) → X𝑥 ∈ 𝐴 ∪ 𝐾 = ∪ 𝐽)
17	14, 16	eqtrd 2770	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐾 ∈ (TopOn‘𝐵)) → X𝑥 ∈ 𝐴 𝐵 = ∪ 𝐽)
18		istopon 22848	. 2 ⊢ (𝐽 ∈ (TopOn‘X𝑥 ∈ 𝐴 𝐵) ↔ (𝐽 ∈ Top ∧ X𝑥 ∈ 𝐴 𝐵 = ∪ 𝐽))
19	9, 17, 18	sylanbrc 583	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐾 ∈ (TopOn‘𝐵)) → 𝐽 ∈ (TopOn‘X𝑥 ∈ 𝐴 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3051 ∪ cuni 4883 ↦ cmpt 5201 ⟶wf 6526 ‘cfv 6530 Xcixp 8909 ∏_tcpt 17450 Topctop 22829 TopOnctopon 22846

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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-om 7860 df-1o 8478 df-2o 8479 df-ixp 8910 df-en 8958 df-fin 8961 df-fi 9421 df-topgen 17455 df-pt 17456 df-top 22830 df-topon 22847 df-bases 22882

This theorem is referenced by: pttoponconst 23533 ptclsg 23551 dfac14lem 23553 ptcnp 23558 prdstps 23565

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