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Mirrors > Home > MPE Home > Th. List > ablfac1lem | Structured version Visualization version GIF version |
Description: Lemma for ablfac1b 19121. 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 3964 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ ℙ) |
4 | prmnn 16006 | . . . . . 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 18840 | . . . . . . . . 9 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
8 | ablfac1.b | . . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺) | |
9 | 8 | grpbn0 18070 | . . . . . . . . 9 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅) |
10 | 6, 7, 9 | 3syl 18 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ≠ ∅) |
11 | ablfac1.f | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ Fin) | |
12 | hashnncl 13715 | . . . . . . . . 9 ⊢ (𝐵 ∈ Fin → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅)) | |
13 | 11, 12 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅)) |
14 | 10, 13 | mpbird 258 | . . . . . . 7 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ) |
15 | 14 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (♯‘𝐵) ∈ ℕ) |
16 | 3, 15 | pccld 16175 | . . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑃 pCnt (♯‘𝐵)) ∈ ℕ0) |
17 | 5, 16 | nnexpcld 13594 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑃↑(𝑃 pCnt (♯‘𝐵))) ∈ ℕ) |
18 | 1, 17 | eqeltrid 2914 | . . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑀 ∈ ℕ) |
19 | ablfac1.n | . . . 4 ⊢ 𝑁 = ((♯‘𝐵) / 𝑀) | |
20 | pcdvds 16188 | . . . . . . 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 5092 | . . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑀 ∥ (♯‘𝐵)) |
23 | nndivdvds 15604 | . . . . . 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 233 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ((♯‘𝐵) / 𝑀) ∈ ℕ) |
26 | 19, 25 | eqeltrid 2914 | . . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑁 ∈ ℕ) |
27 | 18, 26 | jca 512 | . 2 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) |
28 | 1 | oveq1i 7155 | . . 3 ⊢ (𝑀 gcd 𝑁) = ((𝑃↑(𝑃 pCnt (♯‘𝐵))) gcd 𝑁) |
29 | pcndvds2 16192 | . . . . . . 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 7156 | . . . . . . . 8 ⊢ ((♯‘𝐵) / 𝑀) = ((♯‘𝐵) / (𝑃↑(𝑃 pCnt (♯‘𝐵)))) |
32 | 19, 31 | eqtri 2841 | . . . . . . 7 ⊢ 𝑁 = ((♯‘𝐵) / (𝑃↑(𝑃 pCnt (♯‘𝐵)))) |
33 | 32 | breq2i 5065 | . . . . . 6 ⊢ (𝑃 ∥ 𝑁 ↔ 𝑃 ∥ ((♯‘𝐵) / (𝑃↑(𝑃 pCnt (♯‘𝐵))))) |
34 | 30, 33 | sylnibr 330 | . . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ¬ 𝑃 ∥ 𝑁) |
35 | 26 | nnzd 12074 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑁 ∈ ℤ) |
36 | coprm 16043 | . . . . . 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 233 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑃 gcd 𝑁) = 1) |
39 | prmz 16007 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ) | |
40 | 3, 39 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ ℤ) |
41 | rpexp1i 16053 | . . . . 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 1363 | . . . 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 2865 | . 2 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑀 gcd 𝑁) = 1) |
45 | 19 | oveq2i 7156 | . . 3 ⊢ (𝑀 · 𝑁) = (𝑀 · ((♯‘𝐵) / 𝑀)) |
46 | 15 | nncnd 11642 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (♯‘𝐵) ∈ ℂ) |
47 | 18 | nncnd 11642 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑀 ∈ ℂ) |
48 | 18 | nnne0d 11675 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑀 ≠ 0) |
49 | 46, 47, 48 | divcan2d 11406 | . . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑀 · ((♯‘𝐵) / 𝑀)) = (♯‘𝐵)) |
50 | 45, 49 | syl5req 2866 | . 2 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (♯‘𝐵) = (𝑀 · 𝑁)) |
51 | 27, 44, 50 | 3jca 1120 | 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 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 {crab 3139 ⊆ wss 3933 ∅c0 4288 class class class wbr 5057 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7145 Fincfn 8497 1c1 10526 · cmul 10530 / cdiv 11285 ℕcn 11626 ℕ0cn0 11885 ℤcz 11969 ↑cexp 13417 ♯chash 13678 ∥ cdvds 15595 gcd cgcd 15831 ℙcprime 16003 pCnt cpc 16161 Basecbs 16471 Grpcgrp 18041 odcod 18581 Abelcabl 18836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-fz 12881 df-fl 13150 df-mod 13226 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-dvds 15596 df-gcd 15832 df-prm 16004 df-pc 16162 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-abl 18838 |
This theorem is referenced by: ablfac1a 19120 ablfac1b 19121 |
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