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Mirrors > Home > MPE Home > Th. List > gausslemma2dlem0c | Structured version Visualization version GIF version |
Description: Auxiliary lemma 3 for gausslemma2d 25099. (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 3732 | . . . . 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 25082 | . . . . 5 ⊢ (𝜑 → 𝐻 ∈ ℕ) |
6 | 5 | nnnn0d 11351 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ ℕ0) |
7 | 3, 6 | jca 554 | . . 3 ⊢ (𝜑 → (𝑃 ∈ ℙ ∧ 𝐻 ∈ ℕ0)) |
8 | prmnn 15388 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | |
9 | nnre 11027 | . . . . . . . . 9 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℝ) | |
10 | peano2rem 10348 | . . . . . . . . 9 ⊢ (𝑃 ∈ ℝ → (𝑃 − 1) ∈ ℝ) | |
11 | 9, 10 | syl 17 | . . . . . . . 8 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℝ) |
12 | 2re 11090 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ | |
13 | 12 | a1i 11 | . . . . . . . . 9 ⊢ (𝑃 ∈ ℕ → 2 ∈ ℝ) |
14 | 13, 9 | remulcld 10070 | . . . . . . . 8 ⊢ (𝑃 ∈ ℕ → (2 · 𝑃) ∈ ℝ) |
15 | 9 | ltm1d 10956 | . . . . . . . 8 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) < 𝑃) |
16 | nnnn0 11299 | . . . . . . . . . 10 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℕ0) | |
17 | 16 | nn0ge0d 11354 | . . . . . . . . 9 ⊢ (𝑃 ∈ ℕ → 0 ≤ 𝑃) |
18 | 1le2 11241 | . . . . . . . . . 10 ⊢ 1 ≤ 2 | |
19 | 18 | a1i 11 | . . . . . . . . 9 ⊢ (𝑃 ∈ ℕ → 1 ≤ 2) |
20 | 9, 13, 17, 19 | lemulge12d 10962 | . . . . . . . 8 ⊢ (𝑃 ∈ ℕ → 𝑃 ≤ (2 · 𝑃)) |
21 | 11, 9, 14, 15, 20 | ltletrd 10197 | . . . . . . 7 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) < (2 · 𝑃)) |
22 | 2pos 11112 | . . . . . . . . . 10 ⊢ 0 < 2 | |
23 | 12, 22 | pm3.2i 471 | . . . . . . . . 9 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
24 | 23 | a1i 11 | . . . . . . . 8 ⊢ (𝑃 ∈ ℕ → (2 ∈ ℝ ∧ 0 < 2)) |
25 | ltdivmul 10898 | . . . . . . . 8 ⊢ (((𝑃 − 1) ∈ ℝ ∧ 𝑃 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (((𝑃 − 1) / 2) < 𝑃 ↔ (𝑃 − 1) < (2 · 𝑃))) | |
26 | 11, 9, 24, 25 | syl3anc 1326 | . . . . . . 7 ⊢ (𝑃 ∈ ℕ → (((𝑃 − 1) / 2) < 𝑃 ↔ (𝑃 − 1) < (2 · 𝑃))) |
27 | 21, 26 | mpbird 247 | . . . . . 6 ⊢ (𝑃 ∈ ℕ → ((𝑃 − 1) / 2) < 𝑃) |
28 | 2, 8, 27 | 3syl 18 | . . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) < 𝑃) |
29 | 1, 28 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝑃 − 1) / 2) < 𝑃) |
30 | 4, 29 | syl5eqbr 4688 | . . 3 ⊢ (𝜑 → 𝐻 < 𝑃) |
31 | prmndvdsfaclt 15435 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐻 ∈ ℕ0) → (𝐻 < 𝑃 → ¬ 𝑃 ∥ (!‘𝐻))) | |
32 | 7, 30, 31 | sylc 65 | . 2 ⊢ (𝜑 → ¬ 𝑃 ∥ (!‘𝐻)) |
33 | 6 | faccld 13071 | . . . . . 6 ⊢ (𝜑 → (!‘𝐻) ∈ ℕ) |
34 | 33 | nnzd 11481 | . . . . 5 ⊢ (𝜑 → (!‘𝐻) ∈ ℤ) |
35 | nnz 11399 | . . . . . . 7 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℤ) | |
36 | 2, 8, 35 | 3syl 18 | . . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℤ) |
37 | 1, 36 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℤ) |
38 | gcdcom 15235 | . . . . 5 ⊢ (((!‘𝐻) ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((!‘𝐻) gcd 𝑃) = (𝑃 gcd (!‘𝐻))) | |
39 | 34, 37, 38 | syl2anc 693 | . . . 4 ⊢ (𝜑 → ((!‘𝐻) gcd 𝑃) = (𝑃 gcd (!‘𝐻))) |
40 | 39 | eqeq1d 2624 | . . 3 ⊢ (𝜑 → (((!‘𝐻) gcd 𝑃) = 1 ↔ (𝑃 gcd (!‘𝐻)) = 1)) |
41 | coprm 15423 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (!‘𝐻) ∈ ℤ) → (¬ 𝑃 ∥ (!‘𝐻) ↔ (𝑃 gcd (!‘𝐻)) = 1)) | |
42 | 3, 34, 41 | syl2anc 693 | . . 3 ⊢ (𝜑 → (¬ 𝑃 ∥ (!‘𝐻) ↔ (𝑃 gcd (!‘𝐻)) = 1)) |
43 | 40, 42 | bitr4d 271 | . 2 ⊢ (𝜑 → (((!‘𝐻) gcd 𝑃) = 1 ↔ ¬ 𝑃 ∥ (!‘𝐻))) |
44 | 32, 43 | mpbird 247 | 1 ⊢ (𝜑 → ((!‘𝐻) gcd 𝑃) = 1) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∖ cdif 3571 {csn 4177 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 0cc0 9936 1c1 9937 · cmul 9941 < clt 10074 ≤ cle 10075 − cmin 10266 / cdiv 10684 ℕcn 11020 2c2 11070 ℕ0cn0 11292 ℤcz 11377 !cfa 13060 ∥ cdvds 14983 gcd cgcd 15216 ℙcprime 15385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-gcd 15217 df-prm 15386 |
This theorem is referenced by: gausslemma2dlem7 25098 |
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