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Mirrors > Home > MPE Home > Th. List > addmodlteqALT | Structured version Visualization version GIF version |
Description: Two nonnegative integers less than the modulus are equal iff the sums of these integer with another integer are equal modulo the modulus. Shorter proof of addmodlteq 13417 based on the "divides" relation. (Contributed by AV, 14-Mar-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
addmodlteqALT | ⊢ ((𝐼 ∈ (0..^𝑁) ∧ 𝐽 ∈ (0..^𝑁) ∧ 𝑆 ∈ ℤ) → (((𝐼 + 𝑆) mod 𝑁) = ((𝐽 + 𝑆) mod 𝑁) ↔ 𝐼 = 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzo0 13181 | . . . . 5 ⊢ (𝐼 ∈ (0..^𝑁) ↔ (𝐼 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝐼 < 𝑁)) | |
2 | elfzoelz 13141 | . . . . . . . 8 ⊢ (𝐽 ∈ (0..^𝑁) → 𝐽 ∈ ℤ) | |
3 | simplrr 778 | . . . . . . . . . 10 ⊢ (((𝐽 ∈ ℤ ∧ (𝐼 ∈ ℕ0 ∧ 𝑁 ∈ ℕ)) ∧ 𝑆 ∈ ℤ) → 𝑁 ∈ ℕ) | |
4 | nn0z 12098 | . . . . . . . . . . . 12 ⊢ (𝐼 ∈ ℕ0 → 𝐼 ∈ ℤ) | |
5 | 4 | ad2antrl 728 | . . . . . . . . . . 11 ⊢ ((𝐽 ∈ ℤ ∧ (𝐼 ∈ ℕ0 ∧ 𝑁 ∈ ℕ)) → 𝐼 ∈ ℤ) |
6 | zaddcl 12115 | . . . . . . . . . . 11 ⊢ ((𝐼 ∈ ℤ ∧ 𝑆 ∈ ℤ) → (𝐼 + 𝑆) ∈ ℤ) | |
7 | 5, 6 | sylan 583 | . . . . . . . . . 10 ⊢ (((𝐽 ∈ ℤ ∧ (𝐼 ∈ ℕ0 ∧ 𝑁 ∈ ℕ)) ∧ 𝑆 ∈ ℤ) → (𝐼 + 𝑆) ∈ ℤ) |
8 | zaddcl 12115 | . . . . . . . . . . 11 ⊢ ((𝐽 ∈ ℤ ∧ 𝑆 ∈ ℤ) → (𝐽 + 𝑆) ∈ ℤ) | |
9 | 8 | adantlr 715 | . . . . . . . . . 10 ⊢ (((𝐽 ∈ ℤ ∧ (𝐼 ∈ ℕ0 ∧ 𝑁 ∈ ℕ)) ∧ 𝑆 ∈ ℤ) → (𝐽 + 𝑆) ∈ ℤ) |
10 | 3, 7, 9 | 3jca 1129 | . . . . . . . . 9 ⊢ (((𝐽 ∈ ℤ ∧ (𝐼 ∈ ℕ0 ∧ 𝑁 ∈ ℕ)) ∧ 𝑆 ∈ ℤ) → (𝑁 ∈ ℕ ∧ (𝐼 + 𝑆) ∈ ℤ ∧ (𝐽 + 𝑆) ∈ ℤ)) |
11 | 10 | exp31 423 | . . . . . . . 8 ⊢ (𝐽 ∈ ℤ → ((𝐼 ∈ ℕ0 ∧ 𝑁 ∈ ℕ) → (𝑆 ∈ ℤ → (𝑁 ∈ ℕ ∧ (𝐼 + 𝑆) ∈ ℤ ∧ (𝐽 + 𝑆) ∈ ℤ)))) |
12 | 2, 11 | syl 17 | . . . . . . 7 ⊢ (𝐽 ∈ (0..^𝑁) → ((𝐼 ∈ ℕ0 ∧ 𝑁 ∈ ℕ) → (𝑆 ∈ ℤ → (𝑁 ∈ ℕ ∧ (𝐼 + 𝑆) ∈ ℤ ∧ (𝐽 + 𝑆) ∈ ℤ)))) |
13 | 12 | com12 32 | . . . . . 6 ⊢ ((𝐼 ∈ ℕ0 ∧ 𝑁 ∈ ℕ) → (𝐽 ∈ (0..^𝑁) → (𝑆 ∈ ℤ → (𝑁 ∈ ℕ ∧ (𝐼 + 𝑆) ∈ ℤ ∧ (𝐽 + 𝑆) ∈ ℤ)))) |
14 | 13 | 3adant3 1133 | . . . . 5 ⊢ ((𝐼 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝐼 < 𝑁) → (𝐽 ∈ (0..^𝑁) → (𝑆 ∈ ℤ → (𝑁 ∈ ℕ ∧ (𝐼 + 𝑆) ∈ ℤ ∧ (𝐽 + 𝑆) ∈ ℤ)))) |
15 | 1, 14 | sylbi 220 | . . . 4 ⊢ (𝐼 ∈ (0..^𝑁) → (𝐽 ∈ (0..^𝑁) → (𝑆 ∈ ℤ → (𝑁 ∈ ℕ ∧ (𝐼 + 𝑆) ∈ ℤ ∧ (𝐽 + 𝑆) ∈ ℤ)))) |
16 | 15 | 3imp 1112 | . . 3 ⊢ ((𝐼 ∈ (0..^𝑁) ∧ 𝐽 ∈ (0..^𝑁) ∧ 𝑆 ∈ ℤ) → (𝑁 ∈ ℕ ∧ (𝐼 + 𝑆) ∈ ℤ ∧ (𝐽 + 𝑆) ∈ ℤ)) |
17 | moddvds 15722 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝐼 + 𝑆) ∈ ℤ ∧ (𝐽 + 𝑆) ∈ ℤ) → (((𝐼 + 𝑆) mod 𝑁) = ((𝐽 + 𝑆) mod 𝑁) ↔ 𝑁 ∥ ((𝐼 + 𝑆) − (𝐽 + 𝑆)))) | |
18 | 16, 17 | syl 17 | . 2 ⊢ ((𝐼 ∈ (0..^𝑁) ∧ 𝐽 ∈ (0..^𝑁) ∧ 𝑆 ∈ ℤ) → (((𝐼 + 𝑆) mod 𝑁) = ((𝐽 + 𝑆) mod 𝑁) ↔ 𝑁 ∥ ((𝐼 + 𝑆) − (𝐽 + 𝑆)))) |
19 | elfzoel2 13140 | . . . . 5 ⊢ (𝐼 ∈ (0..^𝑁) → 𝑁 ∈ ℤ) | |
20 | zcn 12079 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
21 | 20 | subid1d 11076 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁 − 0) = 𝑁) |
22 | 21 | eqcomd 2745 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 = (𝑁 − 0)) |
23 | 19, 22 | syl 17 | . . . 4 ⊢ (𝐼 ∈ (0..^𝑁) → 𝑁 = (𝑁 − 0)) |
24 | 23 | 3ad2ant1 1134 | . . 3 ⊢ ((𝐼 ∈ (0..^𝑁) ∧ 𝐽 ∈ (0..^𝑁) ∧ 𝑆 ∈ ℤ) → 𝑁 = (𝑁 − 0)) |
25 | elfzoelz 13141 | . . . . 5 ⊢ (𝐼 ∈ (0..^𝑁) → 𝐼 ∈ ℤ) | |
26 | 25 | zcnd 12181 | . . . 4 ⊢ (𝐼 ∈ (0..^𝑁) → 𝐼 ∈ ℂ) |
27 | 2 | zcnd 12181 | . . . 4 ⊢ (𝐽 ∈ (0..^𝑁) → 𝐽 ∈ ℂ) |
28 | zcn 12079 | . . . 4 ⊢ (𝑆 ∈ ℤ → 𝑆 ∈ ℂ) | |
29 | pnpcan2 11016 | . . . 4 ⊢ ((𝐼 ∈ ℂ ∧ 𝐽 ∈ ℂ ∧ 𝑆 ∈ ℂ) → ((𝐼 + 𝑆) − (𝐽 + 𝑆)) = (𝐼 − 𝐽)) | |
30 | 26, 27, 28, 29 | syl3an 1161 | . . 3 ⊢ ((𝐼 ∈ (0..^𝑁) ∧ 𝐽 ∈ (0..^𝑁) ∧ 𝑆 ∈ ℤ) → ((𝐼 + 𝑆) − (𝐽 + 𝑆)) = (𝐼 − 𝐽)) |
31 | 24, 30 | breq12d 5053 | . 2 ⊢ ((𝐼 ∈ (0..^𝑁) ∧ 𝐽 ∈ (0..^𝑁) ∧ 𝑆 ∈ ℤ) → (𝑁 ∥ ((𝐼 + 𝑆) − (𝐽 + 𝑆)) ↔ (𝑁 − 0) ∥ (𝐼 − 𝐽))) |
32 | fzocongeq 15781 | . . 3 ⊢ ((𝐼 ∈ (0..^𝑁) ∧ 𝐽 ∈ (0..^𝑁)) → ((𝑁 − 0) ∥ (𝐼 − 𝐽) ↔ 𝐼 = 𝐽)) | |
33 | 32 | 3adant3 1133 | . 2 ⊢ ((𝐼 ∈ (0..^𝑁) ∧ 𝐽 ∈ (0..^𝑁) ∧ 𝑆 ∈ ℤ) → ((𝑁 − 0) ∥ (𝐼 − 𝐽) ↔ 𝐼 = 𝐽)) |
34 | 18, 31, 33 | 3bitrd 308 | 1 ⊢ ((𝐼 ∈ (0..^𝑁) ∧ 𝐽 ∈ (0..^𝑁) ∧ 𝑆 ∈ ℤ) → (((𝐼 + 𝑆) mod 𝑁) = ((𝐽 + 𝑆) mod 𝑁) ↔ 𝐼 = 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 class class class wbr 5040 (class class class)co 7182 ℂcc 10625 0cc0 10627 + caddc 10630 < clt 10765 − cmin 10960 ℕcn 11728 ℕ0cn0 11988 ℤcz 12074 ..^cfzo 13136 mod cmo 13340 ∥ cdvds 15711 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 ax-pre-sup 10705 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-om 7612 df-1st 7726 df-2nd 7727 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-er 8332 df-en 8568 df-dom 8569 df-sdom 8570 df-sup 8991 df-inf 8992 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-div 11388 df-nn 11729 df-2 11791 df-3 11792 df-n0 11989 df-z 12075 df-uz 12337 df-rp 12485 df-fz 12994 df-fzo 13137 df-fl 13265 df-mod 13341 df-seq 13473 df-exp 13534 df-cj 14560 df-re 14561 df-im 14562 df-sqrt 14696 df-abs 14697 df-dvds 15712 |
This theorem is referenced by: (None) |
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