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Mirrors > Home > MPE Home > Th. List > flbi | Structured version Visualization version GIF version |
Description: A condition equivalent to floor. (Contributed by NM, 11-Mar-2005.) (Revised by Mario Carneiro, 2-Nov-2013.) |
Ref | Expression |
---|---|
flbi | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → ((⌊‘𝐴) = 𝐵 ↔ (𝐵 ≤ 𝐴 ∧ 𝐴 < (𝐵 + 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flval 13156 | . . . 4 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) = (℩𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))) | |
2 | 1 | eqeq1d 2821 | . . 3 ⊢ (𝐴 ∈ ℝ → ((⌊‘𝐴) = 𝐵 ↔ (℩𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))) = 𝐵)) |
3 | 2 | adantr 483 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → ((⌊‘𝐴) = 𝐵 ↔ (℩𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))) = 𝐵)) |
4 | rebtwnz 12339 | . . . 4 ⊢ (𝐴 ∈ ℝ → ∃!𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))) | |
5 | breq1 5060 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥 ≤ 𝐴 ↔ 𝐵 ≤ 𝐴)) | |
6 | oveq1 7155 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥 + 1) = (𝐵 + 1)) | |
7 | 6 | breq2d 5069 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴 < (𝑥 + 1) ↔ 𝐴 < (𝐵 + 1))) |
8 | 5, 7 | anbi12d 632 | . . . . 5 ⊢ (𝑥 = 𝐵 → ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ↔ (𝐵 ≤ 𝐴 ∧ 𝐴 < (𝐵 + 1)))) |
9 | 8 | riota2 7131 | . . . 4 ⊢ ((𝐵 ∈ ℤ ∧ ∃!𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))) → ((𝐵 ≤ 𝐴 ∧ 𝐴 < (𝐵 + 1)) ↔ (℩𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))) = 𝐵)) |
10 | 4, 9 | sylan2 594 | . . 3 ⊢ ((𝐵 ∈ ℤ ∧ 𝐴 ∈ ℝ) → ((𝐵 ≤ 𝐴 ∧ 𝐴 < (𝐵 + 1)) ↔ (℩𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))) = 𝐵)) |
11 | 10 | ancoms 461 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → ((𝐵 ≤ 𝐴 ∧ 𝐴 < (𝐵 + 1)) ↔ (℩𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))) = 𝐵)) |
12 | 3, 11 | bitr4d 284 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → ((⌊‘𝐴) = 𝐵 ↔ (𝐵 ≤ 𝐴 ∧ 𝐴 < (𝐵 + 1)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ∃!wreu 3138 class class class wbr 5057 ‘cfv 6348 ℩crio 7105 (class class class)co 7148 ℝcr 10528 1c1 10530 + caddc 10532 < clt 10667 ≤ cle 10668 ℤcz 11973 ⌊cfl 13152 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-sup 8898 df-inf 8899 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-n0 11890 df-z 11974 df-uz 12236 df-fl 13154 |
This theorem is referenced by: flbi2 13179 fladdz 13187 btwnzge0 13190 bitsfzolem 15775 bitsfzo 15776 bitsmod 15777 bitscmp 15779 pcfaclem 16226 mbfi1fseqlem4 24311 dvfsumlem1 24615 fsumharmonic 25581 ppiub 25772 chpub 25788 bposlem1 25852 bposlem2 25853 ex-fl 28218 subfacval3 32429 itg2addnclem2 34936 hashnzfz2 40644 oddfl 41533 halffl 41553 fourierdlem65 42447 sqrtpwpw2p 43691 flsqrt 43747 fldivexpfllog2 44616 nnlog2ge0lt1 44617 blen1b 44639 |
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