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Mirrors > Home > MPE Home > Th. List > Mathboxes > fltdvdsabdvdsc | Structured version Visualization version GIF version |
Description: Any factor of both 𝐴 and 𝐵 also divides 𝐶. This establishes the validity of fltabcoprmex 40003. (Contributed by SN, 21-Aug-2024.) |
Ref | Expression |
---|---|
fltdvdsabdvdsc.a | ⊢ (𝜑 → 𝐴 ∈ ℕ) |
fltdvdsabdvdsc.b | ⊢ (𝜑 → 𝐵 ∈ ℕ) |
fltdvdsabdvdsc.c | ⊢ (𝜑 → 𝐶 ∈ ℕ) |
fltdvdsabdvdsc.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
fltdvdsabdvdsc.1 | ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁)) |
Ref | Expression |
---|---|
fltdvdsabdvdsc | ⊢ (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fltdvdsabdvdsc.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
2 | fltdvdsabdvdsc.b | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℕ) | |
3 | gcdnncl 15919 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 gcd 𝐵) ∈ ℕ) | |
4 | 1, 2, 3 | syl2anc 587 | . . . . . 6 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∈ ℕ) |
5 | fltdvdsabdvdsc.n | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
6 | 5 | nnnn0d 12007 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
7 | 4, 6 | nnexpcld 13669 | . . . . 5 ⊢ (𝜑 → ((𝐴 gcd 𝐵)↑𝑁) ∈ ℕ) |
8 | 7 | nnzd 12138 | . . . 4 ⊢ (𝜑 → ((𝐴 gcd 𝐵)↑𝑁) ∈ ℤ) |
9 | 1, 6 | nnexpcld 13669 | . . . . 5 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℕ) |
10 | 9 | nnzd 12138 | . . . 4 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℤ) |
11 | 2, 6 | nnexpcld 13669 | . . . . 5 ⊢ (𝜑 → (𝐵↑𝑁) ∈ ℕ) |
12 | 11 | nnzd 12138 | . . . 4 ⊢ (𝜑 → (𝐵↑𝑁) ∈ ℤ) |
13 | 4 | nnzd 12138 | . . . . 5 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∈ ℤ) |
14 | 1 | nnzd 12138 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℤ) |
15 | 2 | nnzd 12138 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℤ) |
16 | gcddvds 15915 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵)) | |
17 | 14, 15, 16 | syl2anc 587 | . . . . . 6 ⊢ (𝜑 → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵)) |
18 | 17 | simpld 498 | . . . . 5 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐴) |
19 | 13, 14, 6, 18 | dvdsexpad 39864 | . . . 4 ⊢ (𝜑 → ((𝐴 gcd 𝐵)↑𝑁) ∥ (𝐴↑𝑁)) |
20 | 17 | simprd 499 | . . . . 5 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐵) |
21 | 13, 15, 6, 20 | dvdsexpad 39864 | . . . 4 ⊢ (𝜑 → ((𝐴 gcd 𝐵)↑𝑁) ∥ (𝐵↑𝑁)) |
22 | 8, 10, 12, 19, 21 | dvds2addd 15706 | . . 3 ⊢ (𝜑 → ((𝐴 gcd 𝐵)↑𝑁) ∥ ((𝐴↑𝑁) + (𝐵↑𝑁))) |
23 | fltdvdsabdvdsc.1 | . . 3 ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁)) | |
24 | 22, 23 | breqtrd 5062 | . 2 ⊢ (𝜑 → ((𝐴 gcd 𝐵)↑𝑁) ∥ (𝐶↑𝑁)) |
25 | fltdvdsabdvdsc.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℕ) | |
26 | dvdsexpnn 39873 | . . 3 ⊢ (((𝐴 gcd 𝐵) ∈ ℕ ∧ 𝐶 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵) ∥ 𝐶 ↔ ((𝐴 gcd 𝐵)↑𝑁) ∥ (𝐶↑𝑁))) | |
27 | 4, 25, 5, 26 | syl3anc 1368 | . 2 ⊢ (𝜑 → ((𝐴 gcd 𝐵) ∥ 𝐶 ↔ ((𝐴 gcd 𝐵)↑𝑁) ∥ (𝐶↑𝑁))) |
28 | 24, 27 | mpbird 260 | 1 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 class class class wbr 5036 (class class class)co 7156 + caddc 10591 ℕcn 11687 ℤcz 12033 ↑cexp 13492 ∥ cdvds 15668 gcd cgcd 15906 |
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 2729 ax-rep 5160 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465 ax-inf2 9150 ax-cnex 10644 ax-resscn 10645 ax-1cn 10646 ax-icn 10647 ax-addcl 10648 ax-addrcl 10649 ax-mulcl 10650 ax-mulrcl 10651 ax-mulcom 10652 ax-addass 10653 ax-mulass 10654 ax-distr 10655 ax-i2m1 10656 ax-1ne0 10657 ax-1rid 10658 ax-rnegex 10659 ax-rrecex 10660 ax-cnre 10661 ax-pre-lttri 10662 ax-pre-lttrn 10663 ax-pre-ltadd 10664 ax-pre-mulgt0 10665 ax-pre-sup 10666 ax-addf 10667 ax-mulf 10668 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-int 4842 df-iun 4888 df-iin 4889 df-br 5037 df-opab 5099 df-mpt 5117 df-tr 5143 df-id 5434 df-eprel 5439 df-po 5447 df-so 5448 df-fr 5487 df-se 5488 df-we 5489 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-pred 6131 df-ord 6177 df-on 6178 df-lim 6179 df-suc 6180 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-isom 6349 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7411 df-om 7586 df-1st 7699 df-2nd 7700 df-supp 7842 df-wrecs 7963 df-recs 8024 df-rdg 8062 df-1o 8118 df-2o 8119 df-er 8305 df-map 8424 df-pm 8425 df-ixp 8493 df-en 8541 df-dom 8542 df-sdom 8543 df-fin 8544 df-fsupp 8880 df-fi 8921 df-sup 8952 df-inf 8953 df-oi 9020 df-card 9414 df-pnf 10728 df-mnf 10729 df-xr 10730 df-ltxr 10731 df-le 10732 df-sub 10923 df-neg 10924 df-div 11349 df-nn 11688 df-2 11750 df-3 11751 df-4 11752 df-5 11753 df-6 11754 df-7 11755 df-8 11756 df-9 11757 df-n0 11948 df-z 12034 df-dec 12151 df-uz 12296 df-q 12402 df-rp 12444 df-xneg 12561 df-xadd 12562 df-xmul 12563 df-ioo 12796 df-ioc 12797 df-ico 12798 df-icc 12799 df-fz 12953 df-fzo 13096 df-fl 13224 df-mod 13300 df-seq 13432 df-exp 13493 df-fac 13697 df-bc 13726 df-hash 13754 df-shft 14487 df-cj 14519 df-re 14520 df-im 14521 df-sqrt 14655 df-abs 14656 df-limsup 14889 df-clim 14906 df-rlim 14907 df-sum 15104 df-ef 15482 df-sin 15484 df-cos 15485 df-pi 15487 df-dvds 15669 df-gcd 15907 df-struct 16556 df-ndx 16557 df-slot 16558 df-base 16560 df-sets 16561 df-ress 16562 df-plusg 16649 df-mulr 16650 df-starv 16651 df-sca 16652 df-vsca 16653 df-ip 16654 df-tset 16655 df-ple 16656 df-ds 16658 df-unif 16659 df-hom 16660 df-cco 16661 df-rest 16767 df-topn 16768 df-0g 16786 df-gsum 16787 df-topgen 16788 df-pt 16789 df-prds 16792 df-xrs 16846 df-qtop 16851 df-imas 16852 df-xps 16854 df-mre 16928 df-mrc 16929 df-acs 16931 df-mgm 17931 df-sgrp 17980 df-mnd 17991 df-submnd 18036 df-mulg 18305 df-cntz 18527 df-cmn 18988 df-psmet 20171 df-xmet 20172 df-met 20173 df-bl 20174 df-mopn 20175 df-fbas 20176 df-fg 20177 df-cnfld 20180 df-top 21607 df-topon 21624 df-topsp 21646 df-bases 21659 df-cld 21732 df-ntr 21733 df-cls 21734 df-nei 21811 df-lp 21849 df-perf 21850 df-cn 21940 df-cnp 21941 df-haus 22028 df-tx 22275 df-hmeo 22468 df-fil 22559 df-fm 22651 df-flim 22652 df-flf 22653 df-xms 23035 df-ms 23036 df-tms 23037 df-cncf 23592 df-limc 24578 df-dv 24579 df-log 25260 df-cxp 25261 |
This theorem is referenced by: flt4lem2 40011 |
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