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Mirrors > Home > MPE Home > Th. List > lcmfdvdsb | Structured version Visualization version GIF version |
Description: Biconditional form of lcmfdvds 16396. (Contributed by AV, 26-Aug-2020.) |
Ref | Expression |
---|---|
lcmfdvdsb | ⊢ ((𝐾 ∈ ℤ ∧ 𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → (∀𝑚 ∈ 𝑍 𝑚 ∥ 𝐾 ↔ (lcm‘𝑍) ∥ 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcmfdvds 16396 | . 2 ⊢ ((𝐾 ∈ ℤ ∧ 𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → (∀𝑚 ∈ 𝑍 𝑚 ∥ 𝐾 → (lcm‘𝑍) ∥ 𝐾)) | |
2 | dvdslcmf 16385 | . . . . . 6 ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → ∀𝑥 ∈ 𝑍 𝑥 ∥ (lcm‘𝑍)) | |
3 | breq1 5084 | . . . . . . . . 9 ⊢ (𝑥 = 𝑚 → (𝑥 ∥ (lcm‘𝑍) ↔ 𝑚 ∥ (lcm‘𝑍))) | |
4 | 3 | rspcv 3562 | . . . . . . . 8 ⊢ (𝑚 ∈ 𝑍 → (∀𝑥 ∈ 𝑍 𝑥 ∥ (lcm‘𝑍) → 𝑚 ∥ (lcm‘𝑍))) |
5 | ssel 3919 | . . . . . . . . . . . . . . . . . 18 ⊢ (𝑍 ⊆ ℤ → (𝑚 ∈ 𝑍 → 𝑚 ∈ ℤ)) | |
6 | 5 | adantr 482 | . . . . . . . . . . . . . . . . 17 ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → (𝑚 ∈ 𝑍 → 𝑚 ∈ ℤ)) |
7 | 6 | com12 32 | . . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ 𝑍 → ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → 𝑚 ∈ ℤ)) |
8 | 7 | adantr 482 | . . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ 𝑍 ∧ 𝐾 ∈ ℤ) → ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → 𝑚 ∈ ℤ)) |
9 | 8 | imp 408 | . . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ 𝑍 ∧ 𝐾 ∈ ℤ) ∧ (𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin)) → 𝑚 ∈ ℤ) |
10 | lcmfcl 16382 | . . . . . . . . . . . . . . . 16 ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → (lcm‘𝑍) ∈ ℕ0) | |
11 | 10 | nn0zd 12474 | . . . . . . . . . . . . . . 15 ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → (lcm‘𝑍) ∈ ℤ) |
12 | 11 | adantl 483 | . . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ 𝑍 ∧ 𝐾 ∈ ℤ) ∧ (𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin)) → (lcm‘𝑍) ∈ ℤ) |
13 | simplr 767 | . . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ 𝑍 ∧ 𝐾 ∈ ℤ) ∧ (𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin)) → 𝐾 ∈ ℤ) | |
14 | dvdstr 16052 | . . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℤ ∧ (lcm‘𝑍) ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑚 ∥ (lcm‘𝑍) ∧ (lcm‘𝑍) ∥ 𝐾) → 𝑚 ∥ 𝐾)) | |
15 | 9, 12, 13, 14 | syl3anc 1371 | . . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ 𝑍 ∧ 𝐾 ∈ ℤ) ∧ (𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin)) → ((𝑚 ∥ (lcm‘𝑍) ∧ (lcm‘𝑍) ∥ 𝐾) → 𝑚 ∥ 𝐾)) |
16 | 15 | expd 417 | . . . . . . . . . . . 12 ⊢ (((𝑚 ∈ 𝑍 ∧ 𝐾 ∈ ℤ) ∧ (𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin)) → (𝑚 ∥ (lcm‘𝑍) → ((lcm‘𝑍) ∥ 𝐾 → 𝑚 ∥ 𝐾))) |
17 | 16 | exp31 421 | . . . . . . . . . . 11 ⊢ (𝑚 ∈ 𝑍 → (𝐾 ∈ ℤ → ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → (𝑚 ∥ (lcm‘𝑍) → ((lcm‘𝑍) ∥ 𝐾 → 𝑚 ∥ 𝐾))))) |
18 | 17 | com23 86 | . . . . . . . . . 10 ⊢ (𝑚 ∈ 𝑍 → ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → (𝐾 ∈ ℤ → (𝑚 ∥ (lcm‘𝑍) → ((lcm‘𝑍) ∥ 𝐾 → 𝑚 ∥ 𝐾))))) |
19 | 18 | com24 95 | . . . . . . . . 9 ⊢ (𝑚 ∈ 𝑍 → (𝑚 ∥ (lcm‘𝑍) → (𝐾 ∈ ℤ → ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → ((lcm‘𝑍) ∥ 𝐾 → 𝑚 ∥ 𝐾))))) |
20 | 19 | com45 97 | . . . . . . . 8 ⊢ (𝑚 ∈ 𝑍 → (𝑚 ∥ (lcm‘𝑍) → (𝐾 ∈ ℤ → ((lcm‘𝑍) ∥ 𝐾 → ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → 𝑚 ∥ 𝐾))))) |
21 | 4, 20 | syld 47 | . . . . . . 7 ⊢ (𝑚 ∈ 𝑍 → (∀𝑥 ∈ 𝑍 𝑥 ∥ (lcm‘𝑍) → (𝐾 ∈ ℤ → ((lcm‘𝑍) ∥ 𝐾 → ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → 𝑚 ∥ 𝐾))))) |
22 | 21 | com15 101 | . . . . . 6 ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → (∀𝑥 ∈ 𝑍 𝑥 ∥ (lcm‘𝑍) → (𝐾 ∈ ℤ → ((lcm‘𝑍) ∥ 𝐾 → (𝑚 ∈ 𝑍 → 𝑚 ∥ 𝐾))))) |
23 | 2, 22 | mpd 15 | . . . . 5 ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → (𝐾 ∈ ℤ → ((lcm‘𝑍) ∥ 𝐾 → (𝑚 ∈ 𝑍 → 𝑚 ∥ 𝐾)))) |
24 | 23 | com12 32 | . . . 4 ⊢ (𝐾 ∈ ℤ → ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → ((lcm‘𝑍) ∥ 𝐾 → (𝑚 ∈ 𝑍 → 𝑚 ∥ 𝐾)))) |
25 | 24 | 3impib 1116 | . . 3 ⊢ ((𝐾 ∈ ℤ ∧ 𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → ((lcm‘𝑍) ∥ 𝐾 → (𝑚 ∈ 𝑍 → 𝑚 ∥ 𝐾))) |
26 | 25 | ralrimdv 3146 | . 2 ⊢ ((𝐾 ∈ ℤ ∧ 𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → ((lcm‘𝑍) ∥ 𝐾 → ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝐾)) |
27 | 1, 26 | impbid 211 | 1 ⊢ ((𝐾 ∈ ℤ ∧ 𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → (∀𝑚 ∈ 𝑍 𝑚 ∥ 𝐾 ↔ (lcm‘𝑍) ∥ 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1087 ∈ wcel 2104 ∀wral 3062 ⊆ wss 3892 class class class wbr 5081 ‘cfv 6458 Fincfn 8764 ℤcz 12369 ∥ cdvds 16012 lcmclcmf 16343 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-inf2 9447 ax-cnex 10977 ax-resscn 10978 ax-1cn 10979 ax-icn 10980 ax-addcl 10981 ax-addrcl 10982 ax-mulcl 10983 ax-mulrcl 10984 ax-mulcom 10985 ax-addass 10986 ax-mulass 10987 ax-distr 10988 ax-i2m1 10989 ax-1ne0 10990 ax-1rid 10991 ax-rnegex 10992 ax-rrecex 10993 ax-cnre 10994 ax-pre-lttri 10995 ax-pre-lttrn 10996 ax-pre-ltadd 10997 ax-pre-mulgt0 10998 ax-pre-sup 10999 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3304 df-reu 3305 df-rab 3306 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-se 5556 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-isom 6467 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-om 7745 df-1st 7863 df-2nd 7864 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-er 8529 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-sup 9249 df-inf 9250 df-oi 9317 df-card 9745 df-pnf 11061 df-mnf 11062 df-xr 11063 df-ltxr 11064 df-le 11065 df-sub 11257 df-neg 11258 df-div 11683 df-nn 12024 df-2 12086 df-3 12087 df-n0 12284 df-z 12370 df-uz 12633 df-rp 12781 df-fz 13290 df-fzo 13433 df-fl 13562 df-mod 13640 df-seq 13772 df-exp 13833 df-hash 14095 df-cj 14859 df-re 14860 df-im 14861 df-sqrt 14995 df-abs 14996 df-clim 15246 df-prod 15665 df-dvds 16013 df-gcd 16251 df-lcm 16344 df-lcmf 16345 |
This theorem is referenced by: aks4d1p3 40286 |
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