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| Mirrors > Home > MPE Home > Th. List > gausslemma2dlem0c | Structured version Visualization version GIF version | ||
| Description: Auxiliary lemma 3 for gausslemma2d 27418. (Contributed by AV, 13-Jul-2021.) | 
| Ref | Expression | 
|---|---|
| gausslemma2dlem0a.p | ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) | 
| gausslemma2dlem0b.h | ⊢ 𝐻 = ((𝑃 − 1) / 2) | 
| Ref | Expression | 
|---|---|
| gausslemma2dlem0c | ⊢ (𝜑 → ((!‘𝐻) gcd 𝑃) = 1) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | gausslemma2dlem0a.p | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) | |
| 2 | eldifi 4131 | . . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ) | |
| 3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑃 ∈ ℙ) | 
| 4 | gausslemma2dlem0b.h | . . . . . 6 ⊢ 𝐻 = ((𝑃 − 1) / 2) | |
| 5 | 1, 4 | gausslemma2dlem0b 27401 | . . . . 5 ⊢ (𝜑 → 𝐻 ∈ ℕ) | 
| 6 | 5 | nnnn0d 12587 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ ℕ0) | 
| 7 | 3, 6 | jca 511 | . . 3 ⊢ (𝜑 → (𝑃 ∈ ℙ ∧ 𝐻 ∈ ℕ0)) | 
| 8 | prmnn 16711 | . . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | |
| 9 | nnre 12273 | . . . . . . . 8 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℝ) | |
| 10 | peano2rem 11576 | . . . . . . . 8 ⊢ (𝑃 ∈ ℝ → (𝑃 − 1) ∈ ℝ) | |
| 11 | 9, 10 | syl 17 | . . . . . . 7 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℝ) | 
| 12 | 2re 12340 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
| 13 | 12 | a1i 11 | . . . . . . . 8 ⊢ (𝑃 ∈ ℕ → 2 ∈ ℝ) | 
| 14 | 13, 9 | remulcld 11291 | . . . . . . 7 ⊢ (𝑃 ∈ ℕ → (2 · 𝑃) ∈ ℝ) | 
| 15 | 9 | ltm1d 12200 | . . . . . . 7 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) < 𝑃) | 
| 16 | nnnn0 12533 | . . . . . . . . 9 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℕ0) | |
| 17 | 16 | nn0ge0d 12590 | . . . . . . . 8 ⊢ (𝑃 ∈ ℕ → 0 ≤ 𝑃) | 
| 18 | 1le2 12475 | . . . . . . . . 9 ⊢ 1 ≤ 2 | |
| 19 | 18 | a1i 11 | . . . . . . . 8 ⊢ (𝑃 ∈ ℕ → 1 ≤ 2) | 
| 20 | 9, 13, 17, 19 | lemulge12d 12206 | . . . . . . 7 ⊢ (𝑃 ∈ ℕ → 𝑃 ≤ (2 · 𝑃)) | 
| 21 | 11, 9, 14, 15, 20 | ltletrd 11421 | . . . . . 6 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) < (2 · 𝑃)) | 
| 22 | 2pos 12369 | . . . . . . . . 9 ⊢ 0 < 2 | |
| 23 | 12, 22 | pm3.2i 470 | . . . . . . . 8 ⊢ (2 ∈ ℝ ∧ 0 < 2) | 
| 24 | 23 | a1i 11 | . . . . . . 7 ⊢ (𝑃 ∈ ℕ → (2 ∈ ℝ ∧ 0 < 2)) | 
| 25 | ltdivmul 12143 | . . . . . . 7 ⊢ (((𝑃 − 1) ∈ ℝ ∧ 𝑃 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (((𝑃 − 1) / 2) < 𝑃 ↔ (𝑃 − 1) < (2 · 𝑃))) | |
| 26 | 11, 9, 24, 25 | syl3anc 1373 | . . . . . 6 ⊢ (𝑃 ∈ ℕ → (((𝑃 − 1) / 2) < 𝑃 ↔ (𝑃 − 1) < (2 · 𝑃))) | 
| 27 | 21, 26 | mpbird 257 | . . . . 5 ⊢ (𝑃 ∈ ℕ → ((𝑃 − 1) / 2) < 𝑃) | 
| 28 | 1, 2, 8, 27 | 4syl 19 | . . . 4 ⊢ (𝜑 → ((𝑃 − 1) / 2) < 𝑃) | 
| 29 | 4, 28 | eqbrtrid 5178 | . . 3 ⊢ (𝜑 → 𝐻 < 𝑃) | 
| 30 | prmndvdsfaclt 16762 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐻 ∈ ℕ0) → (𝐻 < 𝑃 → ¬ 𝑃 ∥ (!‘𝐻))) | |
| 31 | 7, 29, 30 | sylc 65 | . 2 ⊢ (𝜑 → ¬ 𝑃 ∥ (!‘𝐻)) | 
| 32 | 6 | faccld 14323 | . . . . . 6 ⊢ (𝜑 → (!‘𝐻) ∈ ℕ) | 
| 33 | 32 | nnzd 12640 | . . . . 5 ⊢ (𝜑 → (!‘𝐻) ∈ ℤ) | 
| 34 | nnz 12634 | . . . . . 6 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℤ) | |
| 35 | 1, 2, 8, 34 | 4syl 19 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℤ) | 
| 36 | 33, 35 | gcdcomd 16551 | . . . 4 ⊢ (𝜑 → ((!‘𝐻) gcd 𝑃) = (𝑃 gcd (!‘𝐻))) | 
| 37 | 36 | eqeq1d 2739 | . . 3 ⊢ (𝜑 → (((!‘𝐻) gcd 𝑃) = 1 ↔ (𝑃 gcd (!‘𝐻)) = 1)) | 
| 38 | coprm 16748 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (!‘𝐻) ∈ ℤ) → (¬ 𝑃 ∥ (!‘𝐻) ↔ (𝑃 gcd (!‘𝐻)) = 1)) | |
| 39 | 3, 33, 38 | syl2anc 584 | . . 3 ⊢ (𝜑 → (¬ 𝑃 ∥ (!‘𝐻) ↔ (𝑃 gcd (!‘𝐻)) = 1)) | 
| 40 | 37, 39 | bitr4d 282 | . 2 ⊢ (𝜑 → (((!‘𝐻) gcd 𝑃) = 1 ↔ ¬ 𝑃 ∥ (!‘𝐻))) | 
| 41 | 31, 40 | mpbird 257 | 1 ⊢ (𝜑 → ((!‘𝐻) gcd 𝑃) = 1) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 {csn 4626 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 0cc0 11155 1c1 11156 · cmul 11160 < clt 11295 ≤ cle 11296 − cmin 11492 / cdiv 11920 ℕcn 12266 2c2 12321 ℕ0cn0 12526 ℤcz 12613 !cfa 14312 ∥ cdvds 16290 gcd cgcd 16531 ℙcprime 16708 | 
| 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-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-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-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-4 12331 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-dvds 16291 df-gcd 16532 df-prm 16709 | 
| This theorem is referenced by: gausslemma2dlem7 27417 | 
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