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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 3183 | . . 3 ⊢ (𝑅 ∈ (TopOn‘𝑋) → ∀𝑥 ∈ 𝐴 𝑅 ∈ (TopOn‘𝑋)) |
3 | ptuniconst.2 | . . . . 5 ⊢ 𝐽 = (∏t‘(𝐴 × {𝑅})) | |
4 | fconstmpt 5614 | . . . . . 6 ⊢ (𝐴 × {𝑅}) = (𝑥 ∈ 𝐴 ↦ 𝑅) | |
5 | 4 | fveq2i 6673 | . . . . 5 ⊢ (∏t‘(𝐴 × {𝑅})) = (∏t‘(𝑥 ∈ 𝐴 ↦ 𝑅)) |
6 | 3, 5 | eqtri 2844 | . . . 4 ⊢ 𝐽 = (∏t‘(𝑥 ∈ 𝐴 ↦ 𝑅)) |
7 | 6 | pttopon 22204 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘X𝑥 ∈ 𝐴 𝑋)) |
8 | 2, 7 | sylan2 594 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘X𝑥 ∈ 𝐴 𝑋)) |
9 | toponmax 21534 | . . . 4 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝑅) | |
10 | ixpconstg 8470 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝑅) → X𝑥 ∈ 𝐴 𝑋 = (𝑋 ↑m 𝐴)) | |
11 | 9, 10 | sylan2 594 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ (TopOn‘𝑋)) → X𝑥 ∈ 𝐴 𝑋 = (𝑋 ↑m 𝐴)) |
12 | 11 | fveq2d 6674 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ (TopOn‘𝑋)) → (TopOn‘X𝑥 ∈ 𝐴 𝑋) = (TopOn‘(𝑋 ↑m 𝐴))) |
13 | 8, 12 | eleqtrd 2915 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘(𝑋 ↑m 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 {csn 4567 ↦ cmpt 5146 × cxp 5553 ‘cfv 6355 (class class class)co 7156 ↑m cmap 8406 Xcixp 8461 ∏tcpt 16712 TopOnctopon 21518 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-ixp 8462 df-en 8510 df-fin 8513 df-fi 8875 df-topgen 16717 df-pt 16718 df-top 21502 df-topon 21519 df-bases 21554 |
This theorem is referenced by: ptuniconst 22206 pt1hmeo 22414 tmdgsum 22703 efmndtmd 22709 symgtgp 22714 poimir 34940 broucube 34941 |
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