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Mirrors > Home > MPE Home > Th. List > rexuz3 | Structured version Visualization version GIF version |
Description: Restrict the base of the upper integers set to another upper integers set. (Contributed by Mario Carneiro, 26-Dec-2013.) |
Ref | Expression |
---|---|
rexuz3.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
Ref | Expression |
---|---|
rexuz3 | ⊢ (𝑀 ∈ ℤ → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralel 3065 | . . . 4 ⊢ ∀𝑘 ∈ 𝑍 𝑘 ∈ 𝑍 | |
2 | fveq2 6892 | . . . . . . 7 ⊢ (𝑗 = 𝑀 → (ℤ≥‘𝑗) = (ℤ≥‘𝑀)) | |
3 | rexuz3.1 | . . . . . . 7 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
4 | 2, 3 | eqtr4di 2791 | . . . . . 6 ⊢ (𝑗 = 𝑀 → (ℤ≥‘𝑗) = 𝑍) |
5 | 4 | raleqdv 3326 | . . . . 5 ⊢ (𝑗 = 𝑀 → (∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ 𝑍 ↔ ∀𝑘 ∈ 𝑍 𝑘 ∈ 𝑍)) |
6 | 5 | rspcev 3613 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ ∀𝑘 ∈ 𝑍 𝑘 ∈ 𝑍) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ 𝑍) |
7 | 1, 6 | mpan2 690 | . . 3 ⊢ (𝑀 ∈ ℤ → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ 𝑍) |
8 | 7 | biantrurd 534 | . 2 ⊢ (𝑀 ∈ ℤ → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ 𝑍 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑))) |
9 | 3 | uztrn2 12841 | . . . . . . . . . 10 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍) |
10 | 9 | a1d 25 | . . . . . . . . 9 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝜑 → 𝑘 ∈ 𝑍)) |
11 | 10 | ancrd 553 | . . . . . . . 8 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝜑 → (𝑘 ∈ 𝑍 ∧ 𝜑))) |
12 | 11 | ralimdva 3168 | . . . . . . 7 ⊢ (𝑗 ∈ 𝑍 → (∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ 𝑍 ∧ 𝜑))) |
13 | eluzelz 12832 | . . . . . . . 8 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → 𝑗 ∈ ℤ) | |
14 | 13, 3 | eleq2s 2852 | . . . . . . 7 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ) |
15 | 12, 14 | jctild 527 | . . . . . 6 ⊢ (𝑗 ∈ 𝑍 → (∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 → (𝑗 ∈ ℤ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ 𝑍 ∧ 𝜑)))) |
16 | 15 | imp 408 | . . . . 5 ⊢ ((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) → (𝑗 ∈ ℤ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ 𝑍 ∧ 𝜑))) |
17 | uzid 12837 | . . . . . . 7 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ≥‘𝑗)) | |
18 | simpl 484 | . . . . . . . 8 ⊢ ((𝑘 ∈ 𝑍 ∧ 𝜑) → 𝑘 ∈ 𝑍) | |
19 | 18 | ralimi 3084 | . . . . . . 7 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ 𝑍 ∧ 𝜑) → ∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ 𝑍) |
20 | eleq1w 2817 | . . . . . . . 8 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍)) | |
21 | 20 | rspcva 3611 | . . . . . . 7 ⊢ ((𝑗 ∈ (ℤ≥‘𝑗) ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ 𝑍) → 𝑗 ∈ 𝑍) |
22 | 17, 19, 21 | syl2an 597 | . . . . . 6 ⊢ ((𝑗 ∈ ℤ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ 𝑍 ∧ 𝜑)) → 𝑗 ∈ 𝑍) |
23 | simpr 486 | . . . . . . . 8 ⊢ ((𝑘 ∈ 𝑍 ∧ 𝜑) → 𝜑) | |
24 | 23 | ralimi 3084 | . . . . . . 7 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ 𝑍 ∧ 𝜑) → ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) |
25 | 24 | adantl 483 | . . . . . 6 ⊢ ((𝑗 ∈ ℤ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ 𝑍 ∧ 𝜑)) → ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) |
26 | 22, 25 | jca 513 | . . . . 5 ⊢ ((𝑗 ∈ ℤ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ 𝑍 ∧ 𝜑)) → (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑)) |
27 | 16, 26 | impbii 208 | . . . 4 ⊢ ((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) ↔ (𝑗 ∈ ℤ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ 𝑍 ∧ 𝜑))) |
28 | 27 | rexbii2 3091 | . . 3 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ 𝑍 ∧ 𝜑)) |
29 | rexanuz 15292 | . . 3 ⊢ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ 𝑍 ∧ 𝜑) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ 𝑍 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑)) | |
30 | 28, 29 | bitr2i 276 | . 2 ⊢ ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ 𝑍 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) |
31 | 8, 30 | bitr2di 288 | 1 ⊢ (𝑀 ∈ ℤ → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 ‘cfv 6544 ℤcz 12558 ℤ≥cuz 12822 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-pre-lttri 11184 ax-pre-lttrn 11185 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-neg 11447 df-z 12559 df-uz 12823 |
This theorem is referenced by: rexanuz2 15296 cau4 15303 clim2 15448 isercoll 15614 lmbr2 22763 lmff 22805 lmmbr3 24777 iscau3 24795 uniioombllem6 25105 ulmres 25900 rrncmslem 36700 clim2f 44352 clim2f2 44386 climuzlem 44459 lmbr3v 44461 climisp 44462 climrescn 44464 climxrrelem 44465 climxrre 44466 xlimbr 44543 xlimmnfvlem1 44548 xlimmnfvlem2 44549 xlimpnfvlem1 44552 xlimpnfvlem2 44553 |
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