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| Mirrors > Home > MPE Home > Th. List > ablfac1lem | Structured version Visualization version GIF version | ||
| Description: Lemma for ablfac1b 20090. 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 3983 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ ℙ) |
| 4 | prmnn 16711 | . . . . . 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 19803 | . . . . . . . . 9 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
| 8 | ablfac1.b | . . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺) | |
| 9 | 8 | grpbn0 18984 | . . . . . . . . 9 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅) |
| 10 | 6, 7, 9 | 3syl 18 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ≠ ∅) |
| 11 | ablfac1.f | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ Fin) | |
| 12 | hashnncl 14405 | . . . . . . . . 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 16888 | . . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑃 pCnt (♯‘𝐵)) ∈ ℕ0) |
| 17 | 5, 16 | nnexpcld 14284 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑃↑(𝑃 pCnt (♯‘𝐵))) ∈ ℕ) |
| 18 | 1, 17 | eqeltrid 2845 | . . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑀 ∈ ℕ) |
| 19 | ablfac1.n | . . . 4 ⊢ 𝑁 = ((♯‘𝐵) / 𝑀) | |
| 20 | pcdvds 16902 | . . . . . . 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 5178 | . . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑀 ∥ (♯‘𝐵)) |
| 23 | nndivdvds 16299 | . . . . . 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 2845 | . . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑁 ∈ ℕ) |
| 27 | 18, 26 | jca 511 | . 2 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) |
| 28 | 1 | oveq1i 7441 | . . 3 ⊢ (𝑀 gcd 𝑁) = ((𝑃↑(𝑃 pCnt (♯‘𝐵))) gcd 𝑁) |
| 29 | pcndvds2 16906 | . . . . . . 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 7442 | . . . . . . . 8 ⊢ ((♯‘𝐵) / 𝑀) = ((♯‘𝐵) / (𝑃↑(𝑃 pCnt (♯‘𝐵)))) |
| 32 | 19, 31 | eqtri 2765 | . . . . . . 7 ⊢ 𝑁 = ((♯‘𝐵) / (𝑃↑(𝑃 pCnt (♯‘𝐵)))) |
| 33 | 32 | breq2i 5151 | . . . . . 6 ⊢ (𝑃 ∥ 𝑁 ↔ 𝑃 ∥ ((♯‘𝐵) / (𝑃↑(𝑃 pCnt (♯‘𝐵))))) |
| 34 | 30, 33 | sylnibr 329 | . . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ¬ 𝑃 ∥ 𝑁) |
| 35 | 26 | nnzd 12640 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑁 ∈ ℤ) |
| 36 | coprm 16748 | . . . . . 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 16712 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ) | |
| 40 | 3, 39 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ ℤ) |
| 41 | rpexp1i 16760 | . . . . 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 1373 | . . . 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 2789 | . 2 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑀 gcd 𝑁) = 1) |
| 45 | 19 | oveq2i 7442 | . . 3 ⊢ (𝑀 · 𝑁) = (𝑀 · ((♯‘𝐵) / 𝑀)) |
| 46 | 15 | nncnd 12282 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (♯‘𝐵) ∈ ℂ) |
| 47 | 18 | nncnd 12282 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑀 ∈ ℂ) |
| 48 | 18 | nnne0d 12316 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑀 ≠ 0) |
| 49 | 46, 47, 48 | divcan2d 12045 | . . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑀 · ((♯‘𝐵) / 𝑀)) = (♯‘𝐵)) |
| 50 | 45, 49 | eqtr2id 2790 | . 2 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (♯‘𝐵) = (𝑀 · 𝑁)) |
| 51 | 27, 44, 50 | 3jca 1129 | 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 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 {crab 3436 ⊆ wss 3951 ∅c0 4333 class class class wbr 5143 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 1c1 11156 · cmul 11160 / cdiv 11920 ℕcn 12266 ℕ0cn0 12526 ℤcz 12613 ↑cexp 14102 ♯chash 14369 ∥ cdvds 16290 gcd cgcd 16531 ℙcprime 16708 pCnt cpc 16874 Basecbs 17247 Grpcgrp 18951 odcod 19542 Abelcabl 19799 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-fz 13548 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-dvds 16291 df-gcd 16532 df-prm 16709 df-pc 16875 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-abl 19801 |
| This theorem is referenced by: ablfac1a 20089 ablfac1b 20090 |
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