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Mirrors > Home > MPE Home > Th. List > fzocatel | Structured version Visualization version GIF version |
Description: Translate membership in a half-open integer range. (Contributed by Thierry Arnoux, 28-Sep-2018.) |
Ref | Expression |
---|---|
fzocatel | ⊢ (((𝐴 ∈ (0..^(𝐵 + 𝐶)) ∧ ¬ 𝐴 ∈ (0..^𝐵)) ∧ (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (𝐴 − 𝐵) ∈ (0..^𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplr 807 | . . . 4 ⊢ (((𝐴 ∈ (0..^(𝐵 + 𝐶)) ∧ ¬ 𝐴 ∈ (0..^𝐵)) ∧ (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ¬ 𝐴 ∈ (0..^𝐵)) | |
2 | fzospliti 12539 | . . . . . 6 ⊢ ((𝐴 ∈ (0..^(𝐵 + 𝐶)) ∧ 𝐵 ∈ ℤ) → (𝐴 ∈ (0..^𝐵) ∨ 𝐴 ∈ (𝐵..^(𝐵 + 𝐶)))) | |
3 | 2 | ad2ant2r 798 | . . . . 5 ⊢ (((𝐴 ∈ (0..^(𝐵 + 𝐶)) ∧ ¬ 𝐴 ∈ (0..^𝐵)) ∧ (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (𝐴 ∈ (0..^𝐵) ∨ 𝐴 ∈ (𝐵..^(𝐵 + 𝐶)))) |
4 | 3 | ord 391 | . . . 4 ⊢ (((𝐴 ∈ (0..^(𝐵 + 𝐶)) ∧ ¬ 𝐴 ∈ (0..^𝐵)) ∧ (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (¬ 𝐴 ∈ (0..^𝐵) → 𝐴 ∈ (𝐵..^(𝐵 + 𝐶)))) |
5 | 1, 4 | mpd 15 | . . 3 ⊢ (((𝐴 ∈ (0..^(𝐵 + 𝐶)) ∧ ¬ 𝐴 ∈ (0..^𝐵)) ∧ (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 𝐴 ∈ (𝐵..^(𝐵 + 𝐶))) |
6 | simprl 809 | . . 3 ⊢ (((𝐴 ∈ (0..^(𝐵 + 𝐶)) ∧ ¬ 𝐴 ∈ (0..^𝐵)) ∧ (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 𝐵 ∈ ℤ) | |
7 | fzosubel 12566 | . . 3 ⊢ ((𝐴 ∈ (𝐵..^(𝐵 + 𝐶)) ∧ 𝐵 ∈ ℤ) → (𝐴 − 𝐵) ∈ ((𝐵 − 𝐵)..^((𝐵 + 𝐶) − 𝐵))) | |
8 | 5, 6, 7 | syl2anc 694 | . 2 ⊢ (((𝐴 ∈ (0..^(𝐵 + 𝐶)) ∧ ¬ 𝐴 ∈ (0..^𝐵)) ∧ (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (𝐴 − 𝐵) ∈ ((𝐵 − 𝐵)..^((𝐵 + 𝐶) − 𝐵))) |
9 | zcn 11420 | . . . . 5 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℂ) | |
10 | 9 | subidd 10418 | . . . 4 ⊢ (𝐵 ∈ ℤ → (𝐵 − 𝐵) = 0) |
11 | 6, 10 | syl 17 | . . 3 ⊢ (((𝐴 ∈ (0..^(𝐵 + 𝐶)) ∧ ¬ 𝐴 ∈ (0..^𝐵)) ∧ (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (𝐵 − 𝐵) = 0) |
12 | 6 | zcnd 11521 | . . . 4 ⊢ (((𝐴 ∈ (0..^(𝐵 + 𝐶)) ∧ ¬ 𝐴 ∈ (0..^𝐵)) ∧ (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 𝐵 ∈ ℂ) |
13 | simprr 811 | . . . . 5 ⊢ (((𝐴 ∈ (0..^(𝐵 + 𝐶)) ∧ ¬ 𝐴 ∈ (0..^𝐵)) ∧ (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 𝐶 ∈ ℤ) | |
14 | 13 | zcnd 11521 | . . . 4 ⊢ (((𝐴 ∈ (0..^(𝐵 + 𝐶)) ∧ ¬ 𝐴 ∈ (0..^𝐵)) ∧ (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 𝐶 ∈ ℂ) |
15 | 12, 14 | pncan2d 10432 | . . 3 ⊢ (((𝐴 ∈ (0..^(𝐵 + 𝐶)) ∧ ¬ 𝐴 ∈ (0..^𝐵)) ∧ (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((𝐵 + 𝐶) − 𝐵) = 𝐶) |
16 | 11, 15 | oveq12d 6708 | . 2 ⊢ (((𝐴 ∈ (0..^(𝐵 + 𝐶)) ∧ ¬ 𝐴 ∈ (0..^𝐵)) ∧ (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((𝐵 − 𝐵)..^((𝐵 + 𝐶) − 𝐵)) = (0..^𝐶)) |
17 | 8, 16 | eleqtrd 2732 | 1 ⊢ (((𝐴 ∈ (0..^(𝐵 + 𝐶)) ∧ ¬ 𝐴 ∈ (0..^𝐵)) ∧ (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (𝐴 − 𝐵) ∈ (0..^𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 382 ∧ wa 383 = wceq 1523 ∈ wcel 2030 (class class class)co 6690 0cc0 9974 + caddc 9977 − cmin 10304 ℤcz 11415 ..^cfzo 12504 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 df-fzo 12505 |
This theorem is referenced by: ccatcl 13392 repswccat 13578 ofccat 13754 ccatmulgnn0dir 30747 |
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