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Mirrors > Home > MPE Home > Th. List > loclly | Structured version Visualization version GIF version |
Description: If 𝐴 is a local property, then both Locally 𝐴 and 𝑛-Locally 𝐴 simplify to 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.) |
Ref | Expression |
---|---|
loclly | ⊢ (Locally 𝐴 = 𝐴 ↔ 𝑛-Locally 𝐴 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprl 768 | . . . . . . 7 ⊢ ((Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → 𝑗 ∈ 𝐴) | |
2 | simpl 482 | . . . . . . 7 ⊢ ((Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → Locally 𝐴 = 𝐴) | |
3 | 1, 2 | eleqtrrd 2828 | . . . . . 6 ⊢ ((Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → 𝑗 ∈ Locally 𝐴) |
4 | simprr 770 | . . . . . 6 ⊢ ((Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → 𝑥 ∈ 𝑗) | |
5 | llyrest 23311 | . . . . . 6 ⊢ ((𝑗 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝑗) → (𝑗 ↾t 𝑥) ∈ Locally 𝐴) | |
6 | 3, 4, 5 | syl2anc 583 | . . . . 5 ⊢ ((Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ Locally 𝐴) |
7 | 6, 2 | eleqtrd 2827 | . . . 4 ⊢ ((Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝐴) |
8 | 7 | restnlly 23308 | . . 3 ⊢ (Locally 𝐴 = 𝐴 → 𝑛-Locally 𝐴 = Locally 𝐴) |
9 | id 22 | . . 3 ⊢ (Locally 𝐴 = 𝐴 → Locally 𝐴 = 𝐴) | |
10 | 8, 9 | eqtrd 2764 | . 2 ⊢ (Locally 𝐴 = 𝐴 → 𝑛-Locally 𝐴 = 𝐴) |
11 | simprl 768 | . . . . . . 7 ⊢ ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → 𝑗 ∈ 𝐴) | |
12 | simpl 482 | . . . . . . 7 ⊢ ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → 𝑛-Locally 𝐴 = 𝐴) | |
13 | 11, 12 | eleqtrrd 2828 | . . . . . 6 ⊢ ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → 𝑗 ∈ 𝑛-Locally 𝐴) |
14 | simprr 770 | . . . . . 6 ⊢ ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → 𝑥 ∈ 𝑗) | |
15 | nllyrest 23312 | . . . . . 6 ⊢ ((𝑗 ∈ 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗) → (𝑗 ↾t 𝑥) ∈ 𝑛-Locally 𝐴) | |
16 | 13, 14, 15 | syl2anc 583 | . . . . 5 ⊢ ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝑛-Locally 𝐴) |
17 | 16, 12 | eleqtrd 2827 | . . . 4 ⊢ ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝐴) |
18 | 17 | restnlly 23308 | . . 3 ⊢ (𝑛-Locally 𝐴 = 𝐴 → 𝑛-Locally 𝐴 = Locally 𝐴) |
19 | id 22 | . . 3 ⊢ (𝑛-Locally 𝐴 = 𝐴 → 𝑛-Locally 𝐴 = 𝐴) | |
20 | 18, 19 | eqtr3d 2766 | . 2 ⊢ (𝑛-Locally 𝐴 = 𝐴 → Locally 𝐴 = 𝐴) |
21 | 10, 20 | impbii 208 | 1 ⊢ (Locally 𝐴 = 𝐴 ↔ 𝑛-Locally 𝐴 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 (class class class)co 7401 ↾t crest 17365 Locally clly 23290 𝑛-Locally cnlly 23291 |
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 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-en 8936 df-fin 8939 df-fi 9402 df-rest 17367 df-topgen 17388 df-top 22718 df-topon 22735 df-bases 22771 df-nei 22924 df-lly 23292 df-nlly 23293 |
This theorem is referenced by: topnlly 23317 |
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