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Mirrors > Home > MPE Home > Th. List > fzocongeq | Structured version Visualization version GIF version |
Description: Two different elements of a half-open range are not congruent mod its length. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
Ref | Expression |
---|---|
fzocongeq | ⊢ ((𝐴 ∈ (𝐶..^𝐷) ∧ 𝐵 ∈ (𝐶..^𝐷)) → ((𝐷 − 𝐶) ∥ (𝐴 − 𝐵) ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzoel2 13031 | . . . . 5 ⊢ (𝐵 ∈ (𝐶..^𝐷) → 𝐷 ∈ ℤ) | |
2 | elfzoel1 13030 | . . . . 5 ⊢ (𝐵 ∈ (𝐶..^𝐷) → 𝐶 ∈ ℤ) | |
3 | 1, 2 | zsubcld 12086 | . . . 4 ⊢ (𝐵 ∈ (𝐶..^𝐷) → (𝐷 − 𝐶) ∈ ℤ) |
4 | elfzoelz 13032 | . . . . 5 ⊢ (𝐴 ∈ (𝐶..^𝐷) → 𝐴 ∈ ℤ) | |
5 | elfzoelz 13032 | . . . . 5 ⊢ (𝐵 ∈ (𝐶..^𝐷) → 𝐵 ∈ ℤ) | |
6 | zsubcl 12018 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 − 𝐵) ∈ ℤ) | |
7 | 4, 5, 6 | syl2an 597 | . . . 4 ⊢ ((𝐴 ∈ (𝐶..^𝐷) ∧ 𝐵 ∈ (𝐶..^𝐷)) → (𝐴 − 𝐵) ∈ ℤ) |
8 | dvdsabsb 15623 | . . . 4 ⊢ (((𝐷 − 𝐶) ∈ ℤ ∧ (𝐴 − 𝐵) ∈ ℤ) → ((𝐷 − 𝐶) ∥ (𝐴 − 𝐵) ↔ (𝐷 − 𝐶) ∥ (abs‘(𝐴 − 𝐵)))) | |
9 | 3, 7, 8 | syl2an2 684 | . . 3 ⊢ ((𝐴 ∈ (𝐶..^𝐷) ∧ 𝐵 ∈ (𝐶..^𝐷)) → ((𝐷 − 𝐶) ∥ (𝐴 − 𝐵) ↔ (𝐷 − 𝐶) ∥ (abs‘(𝐴 − 𝐵)))) |
10 | fzomaxdif 14697 | . . . 4 ⊢ ((𝐴 ∈ (𝐶..^𝐷) ∧ 𝐵 ∈ (𝐶..^𝐷)) → (abs‘(𝐴 − 𝐵)) ∈ (0..^(𝐷 − 𝐶))) | |
11 | fzo0dvdseq 15667 | . . . 4 ⊢ ((abs‘(𝐴 − 𝐵)) ∈ (0..^(𝐷 − 𝐶)) → ((𝐷 − 𝐶) ∥ (abs‘(𝐴 − 𝐵)) ↔ (abs‘(𝐴 − 𝐵)) = 0)) | |
12 | 10, 11 | syl 17 | . . 3 ⊢ ((𝐴 ∈ (𝐶..^𝐷) ∧ 𝐵 ∈ (𝐶..^𝐷)) → ((𝐷 − 𝐶) ∥ (abs‘(𝐴 − 𝐵)) ↔ (abs‘(𝐴 − 𝐵)) = 0)) |
13 | 9, 12 | bitrd 281 | . 2 ⊢ ((𝐴 ∈ (𝐶..^𝐷) ∧ 𝐵 ∈ (𝐶..^𝐷)) → ((𝐷 − 𝐶) ∥ (𝐴 − 𝐵) ↔ (abs‘(𝐴 − 𝐵)) = 0)) |
14 | 4 | zcnd 12082 | . . . . 5 ⊢ (𝐴 ∈ (𝐶..^𝐷) → 𝐴 ∈ ℂ) |
15 | 5 | zcnd 12082 | . . . . 5 ⊢ (𝐵 ∈ (𝐶..^𝐷) → 𝐵 ∈ ℂ) |
16 | subcl 10879 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) ∈ ℂ) | |
17 | 14, 15, 16 | syl2an 597 | . . . 4 ⊢ ((𝐴 ∈ (𝐶..^𝐷) ∧ 𝐵 ∈ (𝐶..^𝐷)) → (𝐴 − 𝐵) ∈ ℂ) |
18 | 17 | abs00ad 14644 | . . 3 ⊢ ((𝐴 ∈ (𝐶..^𝐷) ∧ 𝐵 ∈ (𝐶..^𝐷)) → ((abs‘(𝐴 − 𝐵)) = 0 ↔ (𝐴 − 𝐵) = 0)) |
19 | subeq0 10906 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵)) | |
20 | 14, 15, 19 | syl2an 597 | . . 3 ⊢ ((𝐴 ∈ (𝐶..^𝐷) ∧ 𝐵 ∈ (𝐶..^𝐷)) → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵)) |
21 | 18, 20 | bitrd 281 | . 2 ⊢ ((𝐴 ∈ (𝐶..^𝐷) ∧ 𝐵 ∈ (𝐶..^𝐷)) → ((abs‘(𝐴 − 𝐵)) = 0 ↔ 𝐴 = 𝐵)) |
22 | 13, 21 | bitrd 281 | 1 ⊢ ((𝐴 ∈ (𝐶..^𝐷) ∧ 𝐵 ∈ (𝐶..^𝐷)) → ((𝐷 − 𝐶) ∥ (𝐴 − 𝐵) ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 0cc0 10531 − cmin 10864 ℤcz 11975 ..^cfzo 13027 abscabs 14587 ∥ cdvds 15601 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-dvds 15602 |
This theorem is referenced by: addmodlteqALT 15669 odf1o2 18692 |
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