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 15565 | . . . . . 6 ⊢ ℤ ≈ ℕ | |
3 | nnct 13350 | . . . . . 6 ⊢ ℕ ≼ ω | |
4 | endomtr 8567 | . . . . . 6 ⊢ ((ℤ ≈ ℕ ∧ ℕ ≼ ω) → ℤ ≼ ω) | |
5 | 2, 3, 4 | mp2an 690 | . . . . 5 ⊢ ℤ ≼ ω |
6 | ovex 7189 | . . . . . . 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 30451 | . . . . 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 5087 | . . 3 ⊢ 𝐼 ≼ ω |
11 | rnct 9947 | . . 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 7476 | . . . . 5 ⊢ (𝑢 × 𝑣) ∈ V |
16 | 15 | rgen2w 3151 | . . . 4 ⊢ ∀𝑢 ∈ ran 𝐼∀𝑣 ∈ ran 𝐼(𝑢 × 𝑣) ∈ V |
17 | 16 | mpocti 30451 | . . 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 5073 | . . . 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 9947 | . . 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 1537 ∈ wcel 2114 Vcvv 3494 class class class wbr 5066 × cxp 5553 ran crn 5556 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 ωcom 7580 ≈ cen 8506 ≼ cdom 8507 1c1 10538 + caddc 10540 / cdiv 11297 ℕcn 11638 2c2 11693 ℤcz 11982 (,)cioo 12739 [,)cico 12741 ↑cexp 13430 topGenctg 16711 |
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 ax-inf2 9104 ax-ac2 9885 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
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-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 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-se 5515 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-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-omul 8107 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-oi 8974 df-card 9368 df-acn 9371 df-ac 9542 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 |
This theorem is referenced by: sxbrsigalem1 31543 |
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