qnegmod - Intuitionistic Logic Explorer

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Theorem qnegmod 10478

Description: The negation of a number modulo a positive number is equal to the difference of the modulus and the number modulo the modulus. (Contributed by Jim Kingdon, 24-Oct-2021.)

**Assertion**
Ref	Expression
qnegmod	⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℚ ∧ 0 < 𝑁) → (-𝐴 mod 𝑁) = ((𝑁 − 𝐴) mod 𝑁))

**Proof of Theorem qnegmod**
Step	Hyp	Ref	Expression
1		qcn 9725	. . . . . 6 ⊢ (𝑁 ∈ ℚ → 𝑁 ∈ ℂ)
2	1	3ad2ant2 1021	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℚ ∧ 0 < 𝑁) → 𝑁 ∈ ℂ)
3		qcn 9725	. . . . . 6 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
4	3	3ad2ant1 1020	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℚ ∧ 0 < 𝑁) → 𝐴 ∈ ℂ)
5	2, 4	negsubd 8360	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℚ ∧ 0 < 𝑁) → (𝑁 + -𝐴) = (𝑁 − 𝐴))
6	5	eqcomd 2202	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℚ ∧ 0 < 𝑁) → (𝑁 − 𝐴) = (𝑁 + -𝐴))
7	6	oveq1d 5940	. 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℚ ∧ 0 < 𝑁) → ((𝑁 − 𝐴) mod 𝑁) = ((𝑁 + -𝐴) mod 𝑁))
8	2	mulid2d 8062	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℚ ∧ 0 < 𝑁) → (1 · 𝑁) = 𝑁)
9	8	oveq1d 5940	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℚ ∧ 0 < 𝑁) → ((1 · 𝑁) + -𝐴) = (𝑁 + -𝐴))
10	9	oveq1d 5940	. 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℚ ∧ 0 < 𝑁) → (((1 · 𝑁) + -𝐴) mod 𝑁) = ((𝑁 + -𝐴) mod 𝑁))
11		1cnd 8059	. . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℚ ∧ 0 < 𝑁) → 1 ∈ ℂ)
12	11, 2	mulcld 8064	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℚ ∧ 0 < 𝑁) → (1 · 𝑁) ∈ ℂ)
13		qnegcl 9727	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → -𝐴 ∈ ℚ)
14		qcn 9725	. . . . . . 7 ⊢ (-𝐴 ∈ ℚ → -𝐴 ∈ ℂ)
15	13, 14	syl 14	. . . . . 6 ⊢ (𝐴 ∈ ℚ → -𝐴 ∈ ℂ)
16	15	3ad2ant1 1020	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℚ ∧ 0 < 𝑁) → -𝐴 ∈ ℂ)
17	12, 16	addcomd 8194	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℚ ∧ 0 < 𝑁) → ((1 · 𝑁) + -𝐴) = (-𝐴 + (1 · 𝑁)))
18	17	oveq1d 5940	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℚ ∧ 0 < 𝑁) → (((1 · 𝑁) + -𝐴) mod 𝑁) = ((-𝐴 + (1 · 𝑁)) mod 𝑁))
19	13	3ad2ant1 1020	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℚ ∧ 0 < 𝑁) → -𝐴 ∈ ℚ)
20		1zzd 9370	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℚ ∧ 0 < 𝑁) → 1 ∈ ℤ)
21		simp2 1000	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℚ ∧ 0 < 𝑁) → 𝑁 ∈ ℚ)
22		simp3 1001	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℚ ∧ 0 < 𝑁) → 0 < 𝑁)
23		modqcyc 10468	. . . 4 ⊢ (((-𝐴 ∈ ℚ ∧ 1 ∈ ℤ) ∧ (𝑁 ∈ ℚ ∧ 0 < 𝑁)) → ((-𝐴 + (1 · 𝑁)) mod 𝑁) = (-𝐴 mod 𝑁))
24	19, 20, 21, 22, 23	syl22anc 1250	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℚ ∧ 0 < 𝑁) → ((-𝐴 + (1 · 𝑁)) mod 𝑁) = (-𝐴 mod 𝑁))
25	18, 24	eqtrd 2229	. 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℚ ∧ 0 < 𝑁) → (((1 · 𝑁) + -𝐴) mod 𝑁) = (-𝐴 mod 𝑁))
26	7, 10, 25	3eqtr2rd 2236	1 ⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℚ ∧ 0 < 𝑁) → (-𝐴 mod 𝑁) = ((𝑁 − 𝐴) mod 𝑁))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ w3a 980 = wceq 1364 ∈ wcel 2167 class class class wbr 4034 (class class class)co 5925 ℂcc 7894 0cc0 7896 1c1 7897 + caddc 7899 · cmul 7901 < clt 8078 − cmin 8214 -cneg 8215 ℤcz 9343 ℚcq 9710 mod cmo 10431

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4152 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-mulrcl 7995 ax-addcom 7996 ax-mulcom 7997 ax-addass 7998 ax-mulass 7999 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-1rid 8003 ax-0id 8004 ax-rnegex 8005 ax-precex 8006 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-apti 8011 ax-pre-ltadd 8012 ax-pre-mulgt0 8013 ax-pre-mulext 8014 ax-arch 8015

This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-id 4329 df-po 4332 df-iso 4333 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-fv 5267 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-reap 8619 df-ap 8626 df-div 8717 df-inn 9008 df-n0 9267 df-z 9344 df-q 9711 df-rp 9746 df-fl 10377 df-mod 10432

This theorem is referenced by: m1modnnsub1 10479 gausslemma2dlem5a 15390

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