![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > lgsneg1 | Structured version Visualization version GIF version |
Description: The Legendre symbol for nonnegative first parameter is unchanged by negation of the second. (Contributed by Mario Carneiro, 4-Feb-2015.) |
Ref | Expression |
---|---|
lgsneg1 | ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ) → (𝐴 /L -𝑁) = (𝐴 /L 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neg0 10648 | . . . 4 ⊢ -0 = 0 | |
2 | simpr 479 | . . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → 𝑁 = 0) | |
3 | 2 | negeqd 10595 | . . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → -𝑁 = -0) |
4 | 1, 3, 2 | 3eqtr4a 2887 | . . 3 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → -𝑁 = 𝑁) |
5 | 4 | oveq2d 6921 | . 2 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → (𝐴 /L -𝑁) = (𝐴 /L 𝑁)) |
6 | nn0z 11728 | . . . . 5 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ) | |
7 | lgsneg 25459 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝐴 /L -𝑁) = (if(𝐴 < 0, -1, 1) · (𝐴 /L 𝑁))) | |
8 | 6, 7 | syl3an1 1206 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝐴 /L -𝑁) = (if(𝐴 < 0, -1, 1) · (𝐴 /L 𝑁))) |
9 | nn0nlt0 11646 | . . . . . . 7 ⊢ (𝐴 ∈ ℕ0 → ¬ 𝐴 < 0) | |
10 | 9 | 3ad2ant1 1167 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → ¬ 𝐴 < 0) |
11 | 10 | iffalsed 4317 | . . . . 5 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → if(𝐴 < 0, -1, 1) = 1) |
12 | 11 | oveq1d 6920 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (if(𝐴 < 0, -1, 1) · (𝐴 /L 𝑁)) = (1 · (𝐴 /L 𝑁))) |
13 | 6 | 3ad2ant1 1167 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 𝐴 ∈ ℤ) |
14 | simp2 1171 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 𝑁 ∈ ℤ) | |
15 | lgscl 25449 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L 𝑁) ∈ ℤ) | |
16 | 13, 14, 15 | syl2anc 579 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝐴 /L 𝑁) ∈ ℤ) |
17 | 16 | zcnd 11811 | . . . . 5 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝐴 /L 𝑁) ∈ ℂ) |
18 | 17 | mulid2d 10375 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (1 · (𝐴 /L 𝑁)) = (𝐴 /L 𝑁)) |
19 | 8, 12, 18 | 3eqtrd 2865 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝐴 /L -𝑁) = (𝐴 /L 𝑁)) |
20 | 19 | 3expa 1151 | . 2 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≠ 0) → (𝐴 /L -𝑁) = (𝐴 /L 𝑁)) |
21 | 5, 20 | pm2.61dane 3086 | 1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ) → (𝐴 /L -𝑁) = (𝐴 /L 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 ifcif 4306 class class class wbr 4873 (class class class)co 6905 0cc0 10252 1c1 10253 · cmul 10257 < clt 10391 -cneg 10586 ℕ0cn0 11618 ℤcz 11704 /L clgs 25432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-2o 7827 df-oadd 7830 df-er 8009 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-sup 8617 df-inf 8618 df-card 9078 df-cda 9305 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-n0 11619 df-xnn0 11691 df-z 11705 df-uz 11969 df-q 12072 df-rp 12113 df-fz 12620 df-fzo 12761 df-fl 12888 df-mod 12964 df-seq 13096 df-exp 13155 df-hash 13411 df-cj 14216 df-re 14217 df-im 14218 df-sqrt 14352 df-abs 14353 df-dvds 15358 df-gcd 15590 df-prm 15758 df-phi 15842 df-pc 15913 df-lgs 25433 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |