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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 16248 | . . . . . 6 ⊢ ℤ ≈ ℕ | |
| 3 | nnct 14022 | . . . . . 6 ⊢ ℕ ≼ ω | |
| 4 | endomtr 9052 | . . . . . 6 ⊢ ((ℤ ≈ ℕ ∧ ℕ ≼ ω) → ℤ ≼ ω) | |
| 5 | 2, 3, 4 | mp2an 692 | . . . . 5 ⊢ ℤ ≼ ω |
| 6 | ovex 7464 | . . . . . . 7 ⊢ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))) ∈ V | |
| 7 | 6 | rgen2w 3066 | . . . . . 6 ⊢ ∀𝑥 ∈ ℤ ∀𝑛 ∈ ℤ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))) ∈ V |
| 8 | 7 | mpocti 32727 | . . . . 5 ⊢ ((ℤ ≼ ω ∧ ℤ ≼ ω) → (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) ≼ ω) |
| 9 | 5, 5, 8 | mp2an 692 | . . . 4 ⊢ (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) ≼ ω |
| 10 | 1, 9 | eqbrtri 5164 | . . 3 ⊢ 𝐼 ≼ ω |
| 11 | rnct 10565 | . . 3 ⊢ (𝐼 ≼ ω → ran 𝐼 ≼ ω) | |
| 12 | 10, 11 | ax-mp 5 | . 2 ⊢ ran 𝐼 ≼ ω |
| 13 | vex 3484 | . . . . . 6 ⊢ 𝑢 ∈ V | |
| 14 | vex 3484 | . . . . . 6 ⊢ 𝑣 ∈ V | |
| 15 | 13, 14 | xpex 7773 | . . . . 5 ⊢ (𝑢 × 𝑣) ∈ V |
| 16 | 15 | rgen2w 3066 | . . . 4 ⊢ ∀𝑢 ∈ ran 𝐼∀𝑣 ∈ ran 𝐼(𝑢 × 𝑣) ∈ V |
| 17 | 16 | mpocti 32727 | . . 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 228 | . . 3 ⊢ ((𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣)) ≼ ω → 𝑅 ≼ ω) |
| 21 | rnct 10565 | . . 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 692 | 1 ⊢ ran 𝑅 ≼ ω |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 × cxp 5683 ran crn 5686 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ωcom 7887 ≈ cen 8982 ≼ cdom 8983 1c1 11156 + caddc 11158 / cdiv 11920 ℕcn 12266 2c2 12321 ℤcz 12613 (,)cioo 13387 [,)cico 13389 ↑cexp 14102 topGenctg 17482 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-ac2 10503 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-omul 8511 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-oi 9550 df-card 9979 df-acn 9982 df-ac 10156 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 |
| This theorem is referenced by: sxbrsigalem1 34287 |
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