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Mirrors > Home > ILE Home > Th. List > gausslemma2dlem0c | GIF version |
Description: Auxiliary lemma 3 for gausslemma2d 15185. (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 3281 | . . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ) | |
3 | 1, 2 | syl 14 | . . . 4 ⊢ (𝜑 → 𝑃 ∈ ℙ) |
4 | gausslemma2dlem0b.h | . . . . . 6 ⊢ 𝐻 = ((𝑃 − 1) / 2) | |
5 | 1, 4 | gausslemma2dlem0b 15166 | . . . . 5 ⊢ (𝜑 → 𝐻 ∈ ℕ) |
6 | 5 | nnnn0d 9293 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ ℕ0) |
7 | 3, 6 | jca 306 | . . 3 ⊢ (𝜑 → (𝑃 ∈ ℙ ∧ 𝐻 ∈ ℕ0)) |
8 | prmnn 12248 | . . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | |
9 | nnre 8989 | . . . . . . . 8 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℝ) | |
10 | peano2rem 8286 | . . . . . . . 8 ⊢ (𝑃 ∈ ℝ → (𝑃 − 1) ∈ ℝ) | |
11 | 9, 10 | syl 14 | . . . . . . 7 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℝ) |
12 | 2re 9052 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
13 | 12 | a1i 9 | . . . . . . . 8 ⊢ (𝑃 ∈ ℕ → 2 ∈ ℝ) |
14 | 13, 9 | remulcld 8050 | . . . . . . 7 ⊢ (𝑃 ∈ ℕ → (2 · 𝑃) ∈ ℝ) |
15 | 9 | ltm1d 8951 | . . . . . . 7 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) < 𝑃) |
16 | nnnn0 9247 | . . . . . . . . 9 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℕ0) | |
17 | 16 | nn0ge0d 9296 | . . . . . . . 8 ⊢ (𝑃 ∈ ℕ → 0 ≤ 𝑃) |
18 | 1le2 9190 | . . . . . . . . 9 ⊢ 1 ≤ 2 | |
19 | 18 | a1i 9 | . . . . . . . 8 ⊢ (𝑃 ∈ ℕ → 1 ≤ 2) |
20 | 9, 13, 17, 19 | lemulge12d 8957 | . . . . . . 7 ⊢ (𝑃 ∈ ℕ → 𝑃 ≤ (2 · 𝑃)) |
21 | 11, 9, 14, 15, 20 | ltletrd 8442 | . . . . . 6 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) < (2 · 𝑃)) |
22 | 2pos 9073 | . . . . . . . . 9 ⊢ 0 < 2 | |
23 | 12, 22 | pm3.2i 272 | . . . . . . . 8 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
24 | 23 | a1i 9 | . . . . . . 7 ⊢ (𝑃 ∈ ℕ → (2 ∈ ℝ ∧ 0 < 2)) |
25 | ltdivmul 8895 | . . . . . . 7 ⊢ (((𝑃 − 1) ∈ ℝ ∧ 𝑃 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (((𝑃 − 1) / 2) < 𝑃 ↔ (𝑃 − 1) < (2 · 𝑃))) | |
26 | 11, 9, 24, 25 | syl3anc 1249 | . . . . . 6 ⊢ (𝑃 ∈ ℕ → (((𝑃 − 1) / 2) < 𝑃 ↔ (𝑃 − 1) < (2 · 𝑃))) |
27 | 21, 26 | mpbird 167 | . . . . 5 ⊢ (𝑃 ∈ ℕ → ((𝑃 − 1) / 2) < 𝑃) |
28 | 1, 2, 8, 27 | 4syl 18 | . . . 4 ⊢ (𝜑 → ((𝑃 − 1) / 2) < 𝑃) |
29 | 4, 28 | eqbrtrid 4064 | . . 3 ⊢ (𝜑 → 𝐻 < 𝑃) |
30 | prmndvdsfaclt 12294 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐻 ∈ ℕ0) → (𝐻 < 𝑃 → ¬ 𝑃 ∥ (!‘𝐻))) | |
31 | 7, 29, 30 | sylc 62 | . 2 ⊢ (𝜑 → ¬ 𝑃 ∥ (!‘𝐻)) |
32 | 6 | faccld 10807 | . . . . . 6 ⊢ (𝜑 → (!‘𝐻) ∈ ℕ) |
33 | 32 | nnzd 9438 | . . . . 5 ⊢ (𝜑 → (!‘𝐻) ∈ ℤ) |
34 | nnz 9336 | . . . . . 6 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℤ) | |
35 | 1, 2, 8, 34 | 4syl 18 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℤ) |
36 | 33, 35 | gcdcomd 12111 | . . . 4 ⊢ (𝜑 → ((!‘𝐻) gcd 𝑃) = (𝑃 gcd (!‘𝐻))) |
37 | 36 | eqeq1d 2202 | . . 3 ⊢ (𝜑 → (((!‘𝐻) gcd 𝑃) = 1 ↔ (𝑃 gcd (!‘𝐻)) = 1)) |
38 | coprm 12282 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (!‘𝐻) ∈ ℤ) → (¬ 𝑃 ∥ (!‘𝐻) ↔ (𝑃 gcd (!‘𝐻)) = 1)) | |
39 | 3, 33, 38 | syl2anc 411 | . . 3 ⊢ (𝜑 → (¬ 𝑃 ∥ (!‘𝐻) ↔ (𝑃 gcd (!‘𝐻)) = 1)) |
40 | 37, 39 | bitr4d 191 | . 2 ⊢ (𝜑 → (((!‘𝐻) gcd 𝑃) = 1 ↔ ¬ 𝑃 ∥ (!‘𝐻))) |
41 | 31, 40 | mpbird 167 | 1 ⊢ (𝜑 → ((!‘𝐻) gcd 𝑃) = 1) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2164 ∖ cdif 3150 {csn 3618 class class class wbr 4029 ‘cfv 5254 (class class class)co 5918 ℝcr 7871 0cc0 7872 1c1 7873 · cmul 7877 < clt 8054 ≤ cle 8055 − cmin 8190 / cdiv 8691 ℕcn 8982 2c2 9033 ℕ0cn0 9240 ℤcz 9317 !cfa 10796 ∥ cdvds 11930 gcd cgcd 12079 ℙcprime 12245 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 ax-arch 7991 ax-caucvg 7992 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-xor 1387 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-frec 6444 df-1o 6469 df-2o 6470 df-er 6587 df-en 6795 df-sup 7043 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-2 9041 df-3 9042 df-4 9043 df-n0 9241 df-z 9318 df-uz 9593 df-q 9685 df-rp 9720 df-fz 10075 df-fzo 10209 df-fl 10339 df-mod 10394 df-seqfrec 10519 df-exp 10610 df-fac 10797 df-cj 10986 df-re 10987 df-im 10988 df-rsqrt 11142 df-abs 11143 df-dvds 11931 df-gcd 12080 df-prm 12246 |
This theorem is referenced by: gausslemma2dlem7 15184 |
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