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 15559 | . . . . . 6 ⊢ ℤ ≈ ℕ | |
3 | nnct 13343 | . . . . . 6 ⊢ ℕ ≼ ω | |
4 | endomtr 8561 | . . . . . 6 ⊢ ((ℤ ≈ ℕ ∧ ℕ ≼ ω) → ℤ ≼ ω) | |
5 | 2, 3, 4 | mp2an 690 | . . . . 5 ⊢ ℤ ≼ ω |
6 | ovex 7183 | . . . . . . 7 ⊢ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))) ∈ V | |
7 | 6 | rgen2w 3151 | . . . . . 6 ⊢ ∀𝑥 ∈ ℤ ∀𝑛 ∈ ℤ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))) ∈ V |
8 | 7 | mpocti 30445 | . . . . 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 5079 | . . 3 ⊢ 𝐼 ≼ ω |
11 | rnct 9941 | . . 3 ⊢ (𝐼 ≼ ω → ran 𝐼 ≼ ω) | |
12 | 10, 11 | ax-mp 5 | . 2 ⊢ ran 𝐼 ≼ ω |
13 | vex 3497 | . . . . . 6 ⊢ 𝑢 ∈ V | |
14 | vex 3497 | . . . . . 6 ⊢ 𝑣 ∈ V | |
15 | 13, 14 | xpex 7470 | . . . . 5 ⊢ (𝑢 × 𝑣) ∈ V |
16 | 15 | rgen2w 3151 | . . . 4 ⊢ ∀𝑢 ∈ ran 𝐼∀𝑣 ∈ ran 𝐼(𝑢 × 𝑣) ∈ V |
17 | 16 | mpocti 30445 | . . 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 5065 | . . . 4 ⊢ (𝑅 ≼ ω ↔ (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣)) ≼ ω) |
20 | 19 | biimpri 230 | . . 3 ⊢ ((𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣)) ≼ ω → 𝑅 ≼ ω) |
21 | rnct 9941 | . . 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 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 class class class wbr 5058 × cxp 5547 ran crn 5550 ‘cfv 6349 (class class class)co 7150 ∈ cmpo 7152 ωcom 7574 ≈ cen 8500 ≼ cdom 8501 1c1 10532 + caddc 10534 / cdiv 11291 ℕcn 11632 2c2 11686 ℤcz 11975 (,)cioo 12732 [,)cico 12734 ↑cexp 13423 topGenctg 16705 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-ac2 9879 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-omul 8101 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-oi 8968 df-card 9362 df-acn 9365 df-ac 9536 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 |
This theorem is referenced by: sxbrsigalem1 31538 |
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