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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvdszzq | Structured version Visualization version GIF version |
Description: Divisibility for an integer quotient. (Contributed by Thierry Arnoux, 17-Sep-2023.) |
Ref | Expression |
---|---|
dvdszzq.1 | ⊢ 𝑁 = (𝐴 / 𝐵) |
dvdszzq.2 | ⊢ (𝜑 → 𝑃 ∈ ℙ) |
dvdszzq.3 | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
dvdszzq.4 | ⊢ (𝜑 → 𝐵 ∈ ℤ) |
dvdszzq.5 | ⊢ (𝜑 → 𝐵 ≠ 0) |
dvdszzq.6 | ⊢ (𝜑 → 𝑃 ∥ 𝐴) |
dvdszzq.7 | ⊢ (𝜑 → ¬ 𝑃 ∥ 𝐵) |
Ref | Expression |
---|---|
dvdszzq | ⊢ (𝜑 → 𝑃 ∥ 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvdszzq.2 | . . 3 ⊢ (𝜑 → 𝑃 ∈ ℙ) | |
2 | dvdszzq.3 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
3 | dvdszzq.4 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℤ) | |
4 | dvdszzq.6 | . . . 4 ⊢ (𝜑 → 𝑃 ∥ 𝐴) | |
5 | dvdszzq.1 | . . . . 5 ⊢ 𝑁 = (𝐴 / 𝐵) | |
6 | 2 | zcnd 12082 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
7 | 3 | zcnd 12082 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
8 | dvdszrcl 15607 | . . . . . . . . 9 ⊢ (𝑃 ∥ 𝐴 → (𝑃 ∈ ℤ ∧ 𝐴 ∈ ℤ)) | |
9 | 8 | simprd 498 | . . . . . . . 8 ⊢ (𝑃 ∥ 𝐴 → 𝐴 ∈ ℤ) |
10 | 4, 9 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℤ) |
11 | 10 | zcnd 12082 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
12 | dvdszzq.5 | . . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 0) | |
13 | 6, 7, 11, 12 | ldiv 11467 | . . . . 5 ⊢ (𝜑 → ((𝑁 · 𝐵) = 𝐴 ↔ 𝑁 = (𝐴 / 𝐵))) |
14 | 5, 13 | mpbiri 260 | . . . 4 ⊢ (𝜑 → (𝑁 · 𝐵) = 𝐴) |
15 | 4, 14 | breqtrrd 5087 | . . 3 ⊢ (𝜑 → 𝑃 ∥ (𝑁 · 𝐵)) |
16 | euclemma 16052 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑃 ∥ (𝑁 · 𝐵) ↔ (𝑃 ∥ 𝑁 ∨ 𝑃 ∥ 𝐵))) | |
17 | 16 | biimpa 479 | . . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝑃 ∥ (𝑁 · 𝐵)) → (𝑃 ∥ 𝑁 ∨ 𝑃 ∥ 𝐵)) |
18 | 1, 2, 3, 15, 17 | syl31anc 1368 | . 2 ⊢ (𝜑 → (𝑃 ∥ 𝑁 ∨ 𝑃 ∥ 𝐵)) |
19 | dvdszzq.7 | . 2 ⊢ (𝜑 → ¬ 𝑃 ∥ 𝐵) | |
20 | orcom 866 | . . 3 ⊢ ((𝑃 ∥ 𝑁 ∨ 𝑃 ∥ 𝐵) ↔ (𝑃 ∥ 𝐵 ∨ 𝑃 ∥ 𝑁)) | |
21 | df-or 844 | . . 3 ⊢ ((𝑃 ∥ 𝐵 ∨ 𝑃 ∥ 𝑁) ↔ (¬ 𝑃 ∥ 𝐵 → 𝑃 ∥ 𝑁)) | |
22 | 20, 21 | sylbb 221 | . 2 ⊢ ((𝑃 ∥ 𝑁 ∨ 𝑃 ∥ 𝐵) → (¬ 𝑃 ∥ 𝐵 → 𝑃 ∥ 𝑁)) |
23 | 18, 19, 22 | sylc 65 | 1 ⊢ (𝜑 → 𝑃 ∥ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 843 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 class class class wbr 5059 (class class class)co 7149 0cc0 10530 · cmul 10535 / cdiv 11290 ℤcz 11975 ∥ cdvds 15602 ℙcprime 16010 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-2o 8096 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-sup 8899 df-inf 8900 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fl 13159 df-mod 13235 df-seq 13367 df-exp 13427 df-cj 14453 df-re 14454 df-im 14455 df-sqrt 14589 df-abs 14590 df-dvds 15603 df-gcd 15839 df-prm 16011 |
This theorem is referenced by: prmdvdsbc 30532 |
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