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Mirrors > Home > MPE Home > Th. List > lgsmodeq | Structured version Visualization version GIF version |
Description: The Legendre (Jacobi) symbol is preserved under reduction mod 𝑛 when 𝑛 is odd. Theorem 9.9(c) in [ApostolNT] p. 188. (Contributed by AV, 20-Jul-2021.) |
Ref | Expression |
---|---|
lgsmodeq | ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ (𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁)) → ((𝐴 mod 𝑁) = (𝐵 mod 𝑁) → (𝐴 /L 𝑁) = (𝐵 /L 𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3anass 1096 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ↔ (𝐴 ∈ ℤ ∧ (𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁))) | |
2 | 1 | biimpri 227 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁)) → (𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁)) |
3 | 2 | 3adant2 1132 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ (𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁)) → (𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁)) |
4 | lgsmod 26674 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) → ((𝐴 mod 𝑁) /L 𝑁) = (𝐴 /L 𝑁)) | |
5 | 3, 4 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ (𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁)) → ((𝐴 mod 𝑁) /L 𝑁) = (𝐴 /L 𝑁)) |
6 | oveq1 7365 | . . . 4 ⊢ ((𝐴 mod 𝑁) = (𝐵 mod 𝑁) → ((𝐴 mod 𝑁) /L 𝑁) = ((𝐵 mod 𝑁) /L 𝑁)) | |
7 | 5, 6 | sylan9req 2798 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ (𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁)) ∧ (𝐴 mod 𝑁) = (𝐵 mod 𝑁)) → (𝐴 /L 𝑁) = ((𝐵 mod 𝑁) /L 𝑁)) |
8 | 3anass 1096 | . . . . . . 7 ⊢ ((𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ↔ (𝐵 ∈ ℤ ∧ (𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁))) | |
9 | 8 | biimpri 227 | . . . . . 6 ⊢ ((𝐵 ∈ ℤ ∧ (𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁)) → (𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁)) |
10 | 9 | 3adant1 1131 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ (𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁)) → (𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁)) |
11 | lgsmod 26674 | . . . . 5 ⊢ ((𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) → ((𝐵 mod 𝑁) /L 𝑁) = (𝐵 /L 𝑁)) | |
12 | 10, 11 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ (𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁)) → ((𝐵 mod 𝑁) /L 𝑁) = (𝐵 /L 𝑁)) |
13 | 12 | adantr 482 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ (𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁)) ∧ (𝐴 mod 𝑁) = (𝐵 mod 𝑁)) → ((𝐵 mod 𝑁) /L 𝑁) = (𝐵 /L 𝑁)) |
14 | 7, 13 | eqtrd 2777 | . 2 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ (𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁)) ∧ (𝐴 mod 𝑁) = (𝐵 mod 𝑁)) → (𝐴 /L 𝑁) = (𝐵 /L 𝑁)) |
15 | 14 | ex 414 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ (𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁)) → ((𝐴 mod 𝑁) = (𝐵 mod 𝑁) → (𝐴 /L 𝑁) = (𝐵 /L 𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 class class class wbr 5106 (class class class)co 7358 ℕcn 12154 2c2 12209 ℤcz 12500 mod cmo 13775 ∥ cdvds 16137 /L clgs 26645 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-pre-sup 11130 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-2o 8414 df-oadd 8417 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-sup 9379 df-inf 9380 df-dju 9838 df-card 9876 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-div 11814 df-nn 12155 df-2 12217 df-3 12218 df-n0 12415 df-xnn0 12487 df-z 12501 df-uz 12765 df-q 12875 df-rp 12917 df-fz 13426 df-fzo 13569 df-fl 13698 df-mod 13776 df-seq 13908 df-exp 13969 df-hash 14232 df-cj 14985 df-re 14986 df-im 14987 df-sqrt 15121 df-abs 15122 df-dvds 16138 df-gcd 16376 df-prm 16549 df-phi 16639 df-pc 16710 df-lgs 26646 |
This theorem is referenced by: (None) |
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