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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fzneg | Structured version Visualization version GIF version |
Description: Reflection of a finite range of integers about 0. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
Ref | Expression |
---|---|
fzneg | ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 ∈ (𝐵...𝐶) ↔ -𝐴 ∈ (-𝐶...-𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ancom 462 | . . 3 ⊢ ((𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐶) ↔ (𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐴)) | |
2 | zre 12559 | . . . . . 6 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ) | |
3 | 2 | 3ad2ant1 1134 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 𝐴 ∈ ℝ) |
4 | zre 12559 | . . . . . 6 ⊢ (𝐶 ∈ ℤ → 𝐶 ∈ ℝ) | |
5 | 4 | 3ad2ant3 1136 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 𝐶 ∈ ℝ) |
6 | 3, 5 | lenegd 11790 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 ≤ 𝐶 ↔ -𝐶 ≤ -𝐴)) |
7 | zre 12559 | . . . . . 6 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ) | |
8 | 7 | 3ad2ant2 1135 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 𝐵 ∈ ℝ) |
9 | 8, 3 | lenegd 11790 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐵 ≤ 𝐴 ↔ -𝐴 ≤ -𝐵)) |
10 | 6, 9 | anbi12d 632 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐴) ↔ (-𝐶 ≤ -𝐴 ∧ -𝐴 ≤ -𝐵))) |
11 | 1, 10 | bitrid 283 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐶) ↔ (-𝐶 ≤ -𝐴 ∧ -𝐴 ≤ -𝐵))) |
12 | elfz 13487 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 ∈ (𝐵...𝐶) ↔ (𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐶))) | |
13 | znegcl 12594 | . . . 4 ⊢ (𝐴 ∈ ℤ → -𝐴 ∈ ℤ) | |
14 | znegcl 12594 | . . . 4 ⊢ (𝐶 ∈ ℤ → -𝐶 ∈ ℤ) | |
15 | znegcl 12594 | . . . 4 ⊢ (𝐵 ∈ ℤ → -𝐵 ∈ ℤ) | |
16 | elfz 13487 | . . . 4 ⊢ ((-𝐴 ∈ ℤ ∧ -𝐶 ∈ ℤ ∧ -𝐵 ∈ ℤ) → (-𝐴 ∈ (-𝐶...-𝐵) ↔ (-𝐶 ≤ -𝐴 ∧ -𝐴 ≤ -𝐵))) | |
17 | 13, 14, 15, 16 | syl3an 1161 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (-𝐴 ∈ (-𝐶...-𝐵) ↔ (-𝐶 ≤ -𝐴 ∧ -𝐴 ≤ -𝐵))) |
18 | 17 | 3com23 1127 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (-𝐴 ∈ (-𝐶...-𝐵) ↔ (-𝐶 ≤ -𝐴 ∧ -𝐴 ≤ -𝐵))) |
19 | 11, 12, 18 | 3bitr4d 311 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 ∈ (𝐵...𝐶) ↔ -𝐴 ∈ (-𝐶...-𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 ∈ wcel 2107 class class class wbr 5148 (class class class)co 7406 ℝcr 11106 ≤ cle 11246 -cneg 11442 ℤcz 12555 ...cfz 13481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-z 12556 df-fz 13482 |
This theorem is referenced by: acongeq 41708 |
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