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Mirrors > Home > MPE Home > Th. List > pttoponconst | Structured version Visualization version GIF version |
Description: The base set for a product topology when all factors are the same. (Contributed by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
ptuniconst.2 | ⊢ 𝐽 = (∏t‘(𝐴 × {𝑅})) |
Ref | Expression |
---|---|
pttoponconst | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘(𝑋 ↑𝑚 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . 4 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ (TopOn‘𝑋)) | |
2 | 1 | ralrimivw 2950 | . . 3 ⊢ (𝑅 ∈ (TopOn‘𝑋) → ∀𝑥 ∈ 𝐴 𝑅 ∈ (TopOn‘𝑋)) |
3 | ptuniconst.2 | . . . . 5 ⊢ 𝐽 = (∏t‘(𝐴 × {𝑅})) | |
4 | fconstmpt 5085 | . . . . . 6 ⊢ (𝐴 × {𝑅}) = (𝑥 ∈ 𝐴 ↦ 𝑅) | |
5 | 4 | fveq2i 6106 | . . . . 5 ⊢ (∏t‘(𝐴 × {𝑅})) = (∏t‘(𝑥 ∈ 𝐴 ↦ 𝑅)) |
6 | 3, 5 | eqtri 2632 | . . . 4 ⊢ 𝐽 = (∏t‘(𝑥 ∈ 𝐴 ↦ 𝑅)) |
7 | 6 | pttopon 21209 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘X𝑥 ∈ 𝐴 𝑋)) |
8 | 2, 7 | sylan2 490 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘X𝑥 ∈ 𝐴 𝑋)) |
9 | toponmax 20543 | . . . 4 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝑅) | |
10 | ixpconstg 7803 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝑅) → X𝑥 ∈ 𝐴 𝑋 = (𝑋 ↑𝑚 𝐴)) | |
11 | 9, 10 | sylan2 490 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ (TopOn‘𝑋)) → X𝑥 ∈ 𝐴 𝑋 = (𝑋 ↑𝑚 𝐴)) |
12 | 11 | fveq2d 6107 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ (TopOn‘𝑋)) → (TopOn‘X𝑥 ∈ 𝐴 𝑋) = (TopOn‘(𝑋 ↑𝑚 𝐴))) |
13 | 8, 12 | eleqtrd 2690 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘(𝑋 ↑𝑚 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {csn 4125 ↦ cmpt 4643 × cxp 5036 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 Xcixp 7794 ∏tcpt 15922 TopOnctopon 20518 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-fin 7845 df-fi 8200 df-topgen 15927 df-pt 15928 df-top 20521 df-bases 20522 df-topon 20523 |
This theorem is referenced by: ptuniconst 21211 pt1hmeo 21419 tmdgsum 21709 symgtgp 21715 poimir 32612 broucube 32613 |
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