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Mirrors > Home > MPE Home > Th. List > dvdstrd | Structured version Visualization version GIF version |
Description: The divides relation is transitive, a deduction version of dvdstr 16328. (Contributed by metakunt, 12-May-2024.) |
Ref | Expression |
---|---|
dvdstrd.1 | ⊢ (𝜑 → 𝐾 ∈ ℤ) |
dvdstrd.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
dvdstrd.3 | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
dvdstrd.4 | ⊢ (𝜑 → 𝐾 ∥ 𝑀) |
dvdstrd.5 | ⊢ (𝜑 → 𝑀 ∥ 𝑁) |
Ref | Expression |
---|---|
dvdstrd | ⊢ (𝜑 → 𝐾 ∥ 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvdstrd.4 | . 2 ⊢ (𝜑 → 𝐾 ∥ 𝑀) | |
2 | dvdstrd.5 | . 2 ⊢ (𝜑 → 𝑀 ∥ 𝑁) | |
3 | dvdstrd.1 | . . 3 ⊢ (𝜑 → 𝐾 ∈ ℤ) | |
4 | dvdstrd.2 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
5 | dvdstrd.3 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
6 | dvdstr 16328 | . . 3 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 ∥ 𝑀 ∧ 𝑀 ∥ 𝑁) → 𝐾 ∥ 𝑁)) | |
7 | 3, 4, 5, 6 | syl3anc 1370 | . 2 ⊢ (𝜑 → ((𝐾 ∥ 𝑀 ∧ 𝑀 ∥ 𝑁) → 𝐾 ∥ 𝑁)) |
8 | 1, 2, 7 | mp2and 699 | 1 ⊢ (𝜑 → 𝐾 ∥ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 class class class wbr 5148 ℤcz 12611 ∥ cdvds 16287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-ltxr 11298 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-dvds 16288 |
This theorem is referenced by: dvdsexp2im 16361 bitsmod 16470 dvdsmulgcd 16590 gcddvdslcm 16636 lcmfunsnlem2lem2 16673 mulgcddvds 16689 rpmulgcd2 16690 rpdvds 16694 isprm5 16741 rpexp 16756 prmdvdsncoprmbd 16761 phimullem 16813 pcpremul 16877 pcdvdstr 16910 pockthlem 16939 4sqlem8 16979 ablfac1eu 20108 znunit 21600 fsumdvdsdiaglem 27241 lgsmod 27382 2sqlem3 27479 2sqlem8 27485 lcmineqlem14 42024 aks4d1p9 42070 unitscyglem2 42178 flt4lem2 42634 dvdsacongtr 42973 jm2.20nn 42986 jm2.27a 42994 jm2.27c 42996 |
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