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Mirrors > Home > MPE Home > Th. List > ablfac1lem | Structured version Visualization version GIF version |
Description: Lemma for ablfac1b 20104. 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 3994 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ ℙ) |
4 | prmnn 16707 | . . . . . 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 19817 | . . . . . . . . 9 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
8 | ablfac1.b | . . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺) | |
9 | 8 | grpbn0 18996 | . . . . . . . . 9 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅) |
10 | 6, 7, 9 | 3syl 18 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ≠ ∅) |
11 | ablfac1.f | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ Fin) | |
12 | hashnncl 14401 | . . . . . . . . 9 ⊢ (𝐵 ∈ Fin → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅)) | |
13 | 11, 12 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅)) |
14 | 10, 13 | mpbird 257 | . . . . . . 7 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ) |
15 | 14 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (♯‘𝐵) ∈ ℕ) |
16 | 3, 15 | pccld 16883 | . . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑃 pCnt (♯‘𝐵)) ∈ ℕ0) |
17 | 5, 16 | nnexpcld 14280 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑃↑(𝑃 pCnt (♯‘𝐵))) ∈ ℕ) |
18 | 1, 17 | eqeltrid 2842 | . . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑀 ∈ ℕ) |
19 | ablfac1.n | . . . 4 ⊢ 𝑁 = ((♯‘𝐵) / 𝑀) | |
20 | pcdvds 16897 | . . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ (♯‘𝐵) ∈ ℕ) → (𝑃↑(𝑃 pCnt (♯‘𝐵))) ∥ (♯‘𝐵)) | |
21 | 3, 15, 20 | syl2anc 584 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑃↑(𝑃 pCnt (♯‘𝐵))) ∥ (♯‘𝐵)) |
22 | 1, 21 | eqbrtrid 5182 | . . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑀 ∥ (♯‘𝐵)) |
23 | nndivdvds 16295 | . . . . . 6 ⊢ (((♯‘𝐵) ∈ ℕ ∧ 𝑀 ∈ ℕ) → (𝑀 ∥ (♯‘𝐵) ↔ ((♯‘𝐵) / 𝑀) ∈ ℕ)) | |
24 | 15, 18, 23 | syl2anc 584 | . . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑀 ∥ (♯‘𝐵) ↔ ((♯‘𝐵) / 𝑀) ∈ ℕ)) |
25 | 22, 24 | mpbid 232 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ((♯‘𝐵) / 𝑀) ∈ ℕ) |
26 | 19, 25 | eqeltrid 2842 | . . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑁 ∈ ℕ) |
27 | 18, 26 | jca 511 | . 2 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) |
28 | 1 | oveq1i 7440 | . . 3 ⊢ (𝑀 gcd 𝑁) = ((𝑃↑(𝑃 pCnt (♯‘𝐵))) gcd 𝑁) |
29 | pcndvds2 16901 | . . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ (♯‘𝐵) ∈ ℕ) → ¬ 𝑃 ∥ ((♯‘𝐵) / (𝑃↑(𝑃 pCnt (♯‘𝐵))))) | |
30 | 3, 15, 29 | syl2anc 584 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ¬ 𝑃 ∥ ((♯‘𝐵) / (𝑃↑(𝑃 pCnt (♯‘𝐵))))) |
31 | 1 | oveq2i 7441 | . . . . . . . 8 ⊢ ((♯‘𝐵) / 𝑀) = ((♯‘𝐵) / (𝑃↑(𝑃 pCnt (♯‘𝐵)))) |
32 | 19, 31 | eqtri 2762 | . . . . . . 7 ⊢ 𝑁 = ((♯‘𝐵) / (𝑃↑(𝑃 pCnt (♯‘𝐵)))) |
33 | 32 | breq2i 5155 | . . . . . 6 ⊢ (𝑃 ∥ 𝑁 ↔ 𝑃 ∥ ((♯‘𝐵) / (𝑃↑(𝑃 pCnt (♯‘𝐵))))) |
34 | 30, 33 | sylnibr 329 | . . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ¬ 𝑃 ∥ 𝑁) |
35 | 26 | nnzd 12637 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑁 ∈ ℤ) |
36 | coprm 16744 | . . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (¬ 𝑃 ∥ 𝑁 ↔ (𝑃 gcd 𝑁) = 1)) | |
37 | 3, 35, 36 | syl2anc 584 | . . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (¬ 𝑃 ∥ 𝑁 ↔ (𝑃 gcd 𝑁) = 1)) |
38 | 34, 37 | mpbid 232 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑃 gcd 𝑁) = 1) |
39 | prmz 16708 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ) | |
40 | 3, 39 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ ℤ) |
41 | rpexp1i 16756 | . . . . 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 1370 | . . . 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 | eqtrid 2786 | . 2 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑀 gcd 𝑁) = 1) |
45 | 19 | oveq2i 7441 | . . 3 ⊢ (𝑀 · 𝑁) = (𝑀 · ((♯‘𝐵) / 𝑀)) |
46 | 15 | nncnd 12279 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (♯‘𝐵) ∈ ℂ) |
47 | 18 | nncnd 12279 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑀 ∈ ℂ) |
48 | 18 | nnne0d 12313 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑀 ≠ 0) |
49 | 46, 47, 48 | divcan2d 12042 | . . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑀 · ((♯‘𝐵) / 𝑀)) = (♯‘𝐵)) |
50 | 45, 49 | eqtr2id 2787 | . 2 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (♯‘𝐵) = (𝑀 · 𝑁)) |
51 | 27, 44, 50 | 3jca 1127 | 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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 {crab 3432 ⊆ wss 3962 ∅c0 4338 class class class wbr 5147 ↦ cmpt 5230 ‘cfv 6562 (class class class)co 7430 Fincfn 8983 1c1 11153 · cmul 11157 / cdiv 11917 ℕcn 12263 ℕ0cn0 12523 ℤcz 12610 ↑cexp 14098 ♯chash 14365 ∥ cdvds 16286 gcd cgcd 16527 ℙcprime 16704 pCnt cpc 16869 Basecbs 17244 Grpcgrp 18963 odcod 19556 Abelcabl 19813 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-inf 9480 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-q 12988 df-rp 13032 df-fz 13544 df-fl 13828 df-mod 13906 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-dvds 16287 df-gcd 16528 df-prm 16705 df-pc 16870 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-abl 19815 |
This theorem is referenced by: ablfac1a 20103 ablfac1b 20104 |
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