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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dya2iocct | Structured version Visualization version GIF version |
Description: The dyadic rectangle set is countable. (Contributed by Thierry Arnoux, 18-Sep-2017.) (Revised by Thierry Arnoux, 11-Oct-2017.) |
Ref | Expression |
---|---|
sxbrsiga.0 | ⊢ 𝐽 = (topGen‘ran (,)) |
dya2ioc.1 | ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) |
dya2ioc.2 | ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣)) |
Ref | Expression |
---|---|
dya2iocct | ⊢ ran 𝑅 ≼ ω |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dya2ioc.1 | . . . 4 ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) | |
2 | znnen 16206 | . . . . . 6 ⊢ ℤ ≈ ℕ | |
3 | nnct 13992 | . . . . . 6 ⊢ ℕ ≼ ω | |
4 | endomtr 9032 | . . . . . 6 ⊢ ((ℤ ≈ ℕ ∧ ℕ ≼ ω) → ℤ ≼ ω) | |
5 | 2, 3, 4 | mp2an 690 | . . . . 5 ⊢ ℤ ≼ ω |
6 | ovex 7446 | . . . . . . 7 ⊢ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))) ∈ V | |
7 | 6 | rgen2w 3056 | . . . . . 6 ⊢ ∀𝑥 ∈ ℤ ∀𝑛 ∈ ℤ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))) ∈ V |
8 | 7 | mpocti 32626 | . . . . 5 ⊢ ((ℤ ≼ ω ∧ ℤ ≼ ω) → (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) ≼ ω) |
9 | 5, 5, 8 | mp2an 690 | . . . 4 ⊢ (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) ≼ ω |
10 | 1, 9 | eqbrtri 5164 | . . 3 ⊢ 𝐼 ≼ ω |
11 | rnct 10556 | . . 3 ⊢ (𝐼 ≼ ω → ran 𝐼 ≼ ω) | |
12 | 10, 11 | ax-mp 5 | . 2 ⊢ ran 𝐼 ≼ ω |
13 | vex 3466 | . . . . . 6 ⊢ 𝑢 ∈ V | |
14 | vex 3466 | . . . . . 6 ⊢ 𝑣 ∈ V | |
15 | 13, 14 | xpex 7750 | . . . . 5 ⊢ (𝑢 × 𝑣) ∈ V |
16 | 15 | rgen2w 3056 | . . . 4 ⊢ ∀𝑢 ∈ ran 𝐼∀𝑣 ∈ ran 𝐼(𝑢 × 𝑣) ∈ V |
17 | 16 | mpocti 32626 | . . 3 ⊢ ((ran 𝐼 ≼ ω ∧ ran 𝐼 ≼ ω) → (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣)) ≼ ω) |
18 | dya2ioc.2 | . . . . 5 ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣)) | |
19 | 18 | breq1i 5150 | . . . 4 ⊢ (𝑅 ≼ ω ↔ (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣)) ≼ ω) |
20 | 19 | biimpri 227 | . . 3 ⊢ ((𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣)) ≼ ω → 𝑅 ≼ ω) |
21 | rnct 10556 | . . 3 ⊢ (𝑅 ≼ ω → ran 𝑅 ≼ ω) | |
22 | 17, 20, 21 | 3syl 18 | . 2 ⊢ ((ran 𝐼 ≼ ω ∧ ran 𝐼 ≼ ω) → ran 𝑅 ≼ ω) |
23 | 12, 12, 22 | mp2an 690 | 1 ⊢ ran 𝑅 ≼ ω |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1534 ∈ wcel 2099 Vcvv 3462 class class class wbr 5143 × cxp 5670 ran crn 5673 ‘cfv 6543 (class class class)co 7413 ∈ cmpo 7415 ωcom 7865 ≈ cen 8960 ≼ cdom 8961 1c1 11147 + caddc 11149 / cdiv 11909 ℕcn 12255 2c2 12310 ℤcz 12601 (,)cioo 13369 [,)cico 13371 ↑cexp 14072 topGenctg 17444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-inf2 9674 ax-ac2 10494 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6302 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7992 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-oadd 8489 df-omul 8490 df-er 8723 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-oi 9543 df-card 9972 df-acn 9975 df-ac 10149 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12256 df-n0 12516 df-z 12602 df-uz 12866 |
This theorem is referenced by: sxbrsigalem1 34129 |
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