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Mirrors > Home > MPE Home > Th. List > cncongrprm | Structured version Visualization version GIF version |
Description: Corollary 2 of Cancellability of Congruences: Two products with a common factor are congruent modulo a prime number not dividing the common factor iff the other factors are congruent modulo the prime number. (Contributed by AV, 13-Jul-2021.) |
Ref | Expression |
---|---|
cncongrprm | ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝑃 ∈ ℙ ∧ ¬ 𝑃 ∥ 𝐶)) → (((𝐴 · 𝐶) mod 𝑃) = ((𝐵 · 𝐶) mod 𝑃) ↔ (𝐴 mod 𝑃) = (𝐵 mod 𝑃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prmnn 16008 | . . . 4 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | |
2 | 1 | ad2antrl 727 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝑃 ∈ ℙ ∧ ¬ 𝑃 ∥ 𝐶)) → 𝑃 ∈ ℕ) |
3 | coprm 16045 | . . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 𝐶 ∈ ℤ) → (¬ 𝑃 ∥ 𝐶 ↔ (𝑃 gcd 𝐶) = 1)) | |
4 | prmz 16009 | . . . . . . . . . . 11 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ) | |
5 | gcdcom 15852 | . . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝑃 gcd 𝐶) = (𝐶 gcd 𝑃)) | |
6 | 4, 5 | sylan 583 | . . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 𝐶 ∈ ℤ) → (𝑃 gcd 𝐶) = (𝐶 gcd 𝑃)) |
7 | 6 | eqeq1d 2800 | . . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 𝐶 ∈ ℤ) → ((𝑃 gcd 𝐶) = 1 ↔ (𝐶 gcd 𝑃) = 1)) |
8 | 3, 7 | bitrd 282 | . . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝐶 ∈ ℤ) → (¬ 𝑃 ∥ 𝐶 ↔ (𝐶 gcd 𝑃) = 1)) |
9 | 8 | ancoms 462 | . . . . . . 7 ⊢ ((𝐶 ∈ ℤ ∧ 𝑃 ∈ ℙ) → (¬ 𝑃 ∥ 𝐶 ↔ (𝐶 gcd 𝑃) = 1)) |
10 | 9 | biimpd 232 | . . . . . 6 ⊢ ((𝐶 ∈ ℤ ∧ 𝑃 ∈ ℙ) → (¬ 𝑃 ∥ 𝐶 → (𝐶 gcd 𝑃) = 1)) |
11 | 10 | expimpd 457 | . . . . 5 ⊢ (𝐶 ∈ ℤ → ((𝑃 ∈ ℙ ∧ ¬ 𝑃 ∥ 𝐶) → (𝐶 gcd 𝑃) = 1)) |
12 | 11 | 3ad2ant3 1132 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝑃 ∈ ℙ ∧ ¬ 𝑃 ∥ 𝐶) → (𝐶 gcd 𝑃) = 1)) |
13 | 12 | imp 410 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝑃 ∈ ℙ ∧ ¬ 𝑃 ∥ 𝐶)) → (𝐶 gcd 𝑃) = 1) |
14 | 2, 13 | jca 515 | . 2 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝑃 ∈ ℙ ∧ ¬ 𝑃 ∥ 𝐶)) → (𝑃 ∈ ℕ ∧ (𝐶 gcd 𝑃) = 1)) |
15 | cncongrcoprm 16004 | . 2 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝑃 ∈ ℕ ∧ (𝐶 gcd 𝑃) = 1)) → (((𝐴 · 𝐶) mod 𝑃) = ((𝐵 · 𝐶) mod 𝑃) ↔ (𝐴 mod 𝑃) = (𝐵 mod 𝑃))) | |
16 | 14, 15 | syldan 594 | 1 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝑃 ∈ ℙ ∧ ¬ 𝑃 ∥ 𝐶)) → (((𝐴 · 𝐶) mod 𝑃) = ((𝐵 · 𝐶) mod 𝑃) ↔ (𝐴 mod 𝑃) = (𝐵 mod 𝑃))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 (class class class)co 7135 1c1 10527 · cmul 10531 ℕcn 11625 ℤcz 11969 mod cmo 13232 ∥ cdvds 15599 gcd cgcd 15833 ℙcprime 16005 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-dvds 15600 df-gcd 15834 df-prm 16006 |
This theorem is referenced by: (None) |
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