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Mirrors > Home > MPE Home > Th. List > ablfac1lem | Structured version Visualization version GIF version |
Description: Lemma for ablfac1b 19192. Satisfy the assumptions of ablfacrp. (Contributed by Mario Carneiro, 26-Apr-2016.) |
Ref | Expression |
---|---|
ablfac1.b | ⊢ 𝐵 = (Base‘𝐺) |
ablfac1.o | ⊢ 𝑂 = (od‘𝐺) |
ablfac1.s | ⊢ 𝑆 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))}) |
ablfac1.g | ⊢ (𝜑 → 𝐺 ∈ Abel) |
ablfac1.f | ⊢ (𝜑 → 𝐵 ∈ Fin) |
ablfac1.1 | ⊢ (𝜑 → 𝐴 ⊆ ℙ) |
ablfac1.m | ⊢ 𝑀 = (𝑃↑(𝑃 pCnt (♯‘𝐵))) |
ablfac1.n | ⊢ 𝑁 = ((♯‘𝐵) / 𝑀) |
Ref | Expression |
---|---|
ablfac1lem | ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑀 gcd 𝑁) = 1 ∧ (♯‘𝐵) = (𝑀 · 𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ablfac1.m | . . . 4 ⊢ 𝑀 = (𝑃↑(𝑃 pCnt (♯‘𝐵))) | |
2 | ablfac1.1 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℙ) | |
3 | 2 | sselda 3967 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ ℙ) |
4 | prmnn 16018 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ ℕ) |
6 | ablfac1.g | . . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ Abel) | |
7 | ablgrp 18911 | . . . . . . . . 9 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
8 | ablfac1.b | . . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺) | |
9 | 8 | grpbn0 18132 | . . . . . . . . 9 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅) |
10 | 6, 7, 9 | 3syl 18 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ≠ ∅) |
11 | ablfac1.f | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ Fin) | |
12 | hashnncl 13728 | . . . . . . . . 9 ⊢ (𝐵 ∈ Fin → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅)) | |
13 | 11, 12 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅)) |
14 | 10, 13 | mpbird 259 | . . . . . . 7 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ) |
15 | 14 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (♯‘𝐵) ∈ ℕ) |
16 | 3, 15 | pccld 16187 | . . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑃 pCnt (♯‘𝐵)) ∈ ℕ0) |
17 | 5, 16 | nnexpcld 13607 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑃↑(𝑃 pCnt (♯‘𝐵))) ∈ ℕ) |
18 | 1, 17 | eqeltrid 2917 | . . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑀 ∈ ℕ) |
19 | ablfac1.n | . . . 4 ⊢ 𝑁 = ((♯‘𝐵) / 𝑀) | |
20 | pcdvds 16200 | . . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ (♯‘𝐵) ∈ ℕ) → (𝑃↑(𝑃 pCnt (♯‘𝐵))) ∥ (♯‘𝐵)) | |
21 | 3, 15, 20 | syl2anc 586 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑃↑(𝑃 pCnt (♯‘𝐵))) ∥ (♯‘𝐵)) |
22 | 1, 21 | eqbrtrid 5101 | . . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑀 ∥ (♯‘𝐵)) |
23 | nndivdvds 15616 | . . . . . 6 ⊢ (((♯‘𝐵) ∈ ℕ ∧ 𝑀 ∈ ℕ) → (𝑀 ∥ (♯‘𝐵) ↔ ((♯‘𝐵) / 𝑀) ∈ ℕ)) | |
24 | 15, 18, 23 | syl2anc 586 | . . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑀 ∥ (♯‘𝐵) ↔ ((♯‘𝐵) / 𝑀) ∈ ℕ)) |
25 | 22, 24 | mpbid 234 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ((♯‘𝐵) / 𝑀) ∈ ℕ) |
26 | 19, 25 | eqeltrid 2917 | . . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑁 ∈ ℕ) |
27 | 18, 26 | jca 514 | . 2 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) |
28 | 1 | oveq1i 7166 | . . 3 ⊢ (𝑀 gcd 𝑁) = ((𝑃↑(𝑃 pCnt (♯‘𝐵))) gcd 𝑁) |
29 | pcndvds2 16204 | . . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ (♯‘𝐵) ∈ ℕ) → ¬ 𝑃 ∥ ((♯‘𝐵) / (𝑃↑(𝑃 pCnt (♯‘𝐵))))) | |
30 | 3, 15, 29 | syl2anc 586 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ¬ 𝑃 ∥ ((♯‘𝐵) / (𝑃↑(𝑃 pCnt (♯‘𝐵))))) |
31 | 1 | oveq2i 7167 | . . . . . . . 8 ⊢ ((♯‘𝐵) / 𝑀) = ((♯‘𝐵) / (𝑃↑(𝑃 pCnt (♯‘𝐵)))) |
32 | 19, 31 | eqtri 2844 | . . . . . . 7 ⊢ 𝑁 = ((♯‘𝐵) / (𝑃↑(𝑃 pCnt (♯‘𝐵)))) |
33 | 32 | breq2i 5074 | . . . . . 6 ⊢ (𝑃 ∥ 𝑁 ↔ 𝑃 ∥ ((♯‘𝐵) / (𝑃↑(𝑃 pCnt (♯‘𝐵))))) |
34 | 30, 33 | sylnibr 331 | . . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ¬ 𝑃 ∥ 𝑁) |
35 | 26 | nnzd 12087 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑁 ∈ ℤ) |
36 | coprm 16055 | . . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (¬ 𝑃 ∥ 𝑁 ↔ (𝑃 gcd 𝑁) = 1)) | |
37 | 3, 35, 36 | syl2anc 586 | . . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (¬ 𝑃 ∥ 𝑁 ↔ (𝑃 gcd 𝑁) = 1)) |
38 | 34, 37 | mpbid 234 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑃 gcd 𝑁) = 1) |
39 | prmz 16019 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ) | |
40 | 3, 39 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ ℤ) |
41 | rpexp1i 16065 | . . . . 5 ⊢ ((𝑃 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑃 pCnt (♯‘𝐵)) ∈ ℕ0) → ((𝑃 gcd 𝑁) = 1 → ((𝑃↑(𝑃 pCnt (♯‘𝐵))) gcd 𝑁) = 1)) | |
42 | 40, 35, 16, 41 | syl3anc 1367 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ((𝑃 gcd 𝑁) = 1 → ((𝑃↑(𝑃 pCnt (♯‘𝐵))) gcd 𝑁) = 1)) |
43 | 38, 42 | mpd 15 | . . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ((𝑃↑(𝑃 pCnt (♯‘𝐵))) gcd 𝑁) = 1) |
44 | 28, 43 | syl5eq 2868 | . 2 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑀 gcd 𝑁) = 1) |
45 | 19 | oveq2i 7167 | . . 3 ⊢ (𝑀 · 𝑁) = (𝑀 · ((♯‘𝐵) / 𝑀)) |
46 | 15 | nncnd 11654 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (♯‘𝐵) ∈ ℂ) |
47 | 18 | nncnd 11654 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑀 ∈ ℂ) |
48 | 18 | nnne0d 11688 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑀 ≠ 0) |
49 | 46, 47, 48 | divcan2d 11418 | . . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑀 · ((♯‘𝐵) / 𝑀)) = (♯‘𝐵)) |
50 | 45, 49 | syl5req 2869 | . 2 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (♯‘𝐵) = (𝑀 · 𝑁)) |
51 | 27, 44, 50 | 3jca 1124 | 1 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑀 gcd 𝑁) = 1 ∧ (♯‘𝐵) = (𝑀 · 𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 {crab 3142 ⊆ wss 3936 ∅c0 4291 class class class wbr 5066 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 Fincfn 8509 1c1 10538 · cmul 10542 / cdiv 11297 ℕcn 11638 ℕ0cn0 11898 ℤcz 11982 ↑cexp 13430 ♯chash 13691 ∥ cdvds 15607 gcd cgcd 15843 ℙcprime 16015 pCnt cpc 16173 Basecbs 16483 Grpcgrp 18103 odcod 18652 Abelcabl 18907 |
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-int 4877 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-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-card 9368 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-q 12350 df-rp 12391 df-fz 12894 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-dvds 15608 df-gcd 15844 df-prm 16016 df-pc 16174 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-abl 18909 |
This theorem is referenced by: ablfac1a 19191 ablfac1b 19192 |
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