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Mirrors > Home > MPE Home > Th. List > divalglem0 | Structured version Visualization version GIF version |
Description: Lemma for divalg 15754. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
divalglem0.1 | ⊢ 𝑁 ∈ ℤ |
divalglem0.2 | ⊢ 𝐷 ∈ ℤ |
Ref | Expression |
---|---|
divalglem0 | ⊢ ((𝑅 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐷 ∥ (𝑁 − 𝑅) → 𝐷 ∥ (𝑁 − (𝑅 − (𝐾 · (abs‘𝐷)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divalglem0.2 | . . . . . 6 ⊢ 𝐷 ∈ ℤ | |
2 | iddvds 15623 | . . . . . . 7 ⊢ (𝐷 ∈ ℤ → 𝐷 ∥ 𝐷) | |
3 | dvdsabsb 15629 | . . . . . . . 8 ⊢ ((𝐷 ∈ ℤ ∧ 𝐷 ∈ ℤ) → (𝐷 ∥ 𝐷 ↔ 𝐷 ∥ (abs‘𝐷))) | |
4 | 3 | anidms 569 | . . . . . . 7 ⊢ (𝐷 ∈ ℤ → (𝐷 ∥ 𝐷 ↔ 𝐷 ∥ (abs‘𝐷))) |
5 | 2, 4 | mpbid 234 | . . . . . 6 ⊢ (𝐷 ∈ ℤ → 𝐷 ∥ (abs‘𝐷)) |
6 | 1, 5 | ax-mp 5 | . . . . 5 ⊢ 𝐷 ∥ (abs‘𝐷) |
7 | nn0abscl 14672 | . . . . . . . 8 ⊢ (𝐷 ∈ ℤ → (abs‘𝐷) ∈ ℕ0) | |
8 | 1, 7 | ax-mp 5 | . . . . . . 7 ⊢ (abs‘𝐷) ∈ ℕ0 |
9 | 8 | nn0zi 12008 | . . . . . 6 ⊢ (abs‘𝐷) ∈ ℤ |
10 | dvdsmultr2 15649 | . . . . . 6 ⊢ ((𝐷 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ (abs‘𝐷) ∈ ℤ) → (𝐷 ∥ (abs‘𝐷) → 𝐷 ∥ (𝐾 · (abs‘𝐷)))) | |
11 | 1, 9, 10 | mp3an13 1448 | . . . . 5 ⊢ (𝐾 ∈ ℤ → (𝐷 ∥ (abs‘𝐷) → 𝐷 ∥ (𝐾 · (abs‘𝐷)))) |
12 | 6, 11 | mpi 20 | . . . 4 ⊢ (𝐾 ∈ ℤ → 𝐷 ∥ (𝐾 · (abs‘𝐷))) |
13 | 12 | adantl 484 | . . 3 ⊢ ((𝑅 ∈ ℤ ∧ 𝐾 ∈ ℤ) → 𝐷 ∥ (𝐾 · (abs‘𝐷))) |
14 | divalglem0.1 | . . . . 5 ⊢ 𝑁 ∈ ℤ | |
15 | zsubcl 12025 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ) → (𝑁 − 𝑅) ∈ ℤ) | |
16 | 14, 15 | mpan 688 | . . . 4 ⊢ (𝑅 ∈ ℤ → (𝑁 − 𝑅) ∈ ℤ) |
17 | zmulcl 12032 | . . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ (abs‘𝐷) ∈ ℤ) → (𝐾 · (abs‘𝐷)) ∈ ℤ) | |
18 | 9, 17 | mpan2 689 | . . . 4 ⊢ (𝐾 ∈ ℤ → (𝐾 · (abs‘𝐷)) ∈ ℤ) |
19 | dvds2add 15643 | . . . 4 ⊢ ((𝐷 ∈ ℤ ∧ (𝑁 − 𝑅) ∈ ℤ ∧ (𝐾 · (abs‘𝐷)) ∈ ℤ) → ((𝐷 ∥ (𝑁 − 𝑅) ∧ 𝐷 ∥ (𝐾 · (abs‘𝐷))) → 𝐷 ∥ ((𝑁 − 𝑅) + (𝐾 · (abs‘𝐷))))) | |
20 | 1, 16, 18, 19 | mp3an3an 1463 | . . 3 ⊢ ((𝑅 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝐷 ∥ (𝑁 − 𝑅) ∧ 𝐷 ∥ (𝐾 · (abs‘𝐷))) → 𝐷 ∥ ((𝑁 − 𝑅) + (𝐾 · (abs‘𝐷))))) |
21 | 13, 20 | mpan2d 692 | . 2 ⊢ ((𝑅 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐷 ∥ (𝑁 − 𝑅) → 𝐷 ∥ ((𝑁 − 𝑅) + (𝐾 · (abs‘𝐷))))) |
22 | zcn 11987 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
23 | 14, 22 | ax-mp 5 | . . . 4 ⊢ 𝑁 ∈ ℂ |
24 | zcn 11987 | . . . 4 ⊢ (𝑅 ∈ ℤ → 𝑅 ∈ ℂ) | |
25 | 18 | zcnd 12089 | . . . 4 ⊢ (𝐾 ∈ ℤ → (𝐾 · (abs‘𝐷)) ∈ ℂ) |
26 | subsub 10916 | . . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (𝐾 · (abs‘𝐷)) ∈ ℂ) → (𝑁 − (𝑅 − (𝐾 · (abs‘𝐷)))) = ((𝑁 − 𝑅) + (𝐾 · (abs‘𝐷)))) | |
27 | 23, 24, 25, 26 | mp3an3an 1463 | . . 3 ⊢ ((𝑅 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑁 − (𝑅 − (𝐾 · (abs‘𝐷)))) = ((𝑁 − 𝑅) + (𝐾 · (abs‘𝐷)))) |
28 | 27 | breq2d 5078 | . 2 ⊢ ((𝑅 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐷 ∥ (𝑁 − (𝑅 − (𝐾 · (abs‘𝐷)))) ↔ 𝐷 ∥ ((𝑁 − 𝑅) + (𝐾 · (abs‘𝐷))))) |
29 | 21, 28 | sylibrd 261 | 1 ⊢ ((𝑅 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐷 ∥ (𝑁 − 𝑅) → 𝐷 ∥ (𝑁 − (𝑅 − (𝐾 · (abs‘𝐷)))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 + caddc 10540 · cmul 10542 − cmin 10870 ℕ0cn0 11898 ℤcz 11982 abscabs 14593 ∥ cdvds 15607 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-dvds 15608 |
This theorem is referenced by: divalglem5 15748 |
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