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Mirrors > Home > MPE Home > Th. List > gausslemma2dlem0c | Structured version Visualization version GIF version |
Description: Auxiliary lemma 3 for gausslemma2d 24899. (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 3694 | . . . . 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 24882 | . . . . 5 ⊢ (𝜑 → 𝐻 ∈ ℕ) |
6 | 5 | nnnn0d 11228 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ ℕ0) |
7 | 3, 6 | jca 553 | . . 3 ⊢ (𝜑 → (𝑃 ∈ ℙ ∧ 𝐻 ∈ ℕ0)) |
8 | prmnn 15226 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | |
9 | nnre 10904 | . . . . . . . . 9 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℝ) | |
10 | peano2rem 10227 | . . . . . . . . 9 ⊢ (𝑃 ∈ ℝ → (𝑃 − 1) ∈ ℝ) | |
11 | 9, 10 | syl 17 | . . . . . . . 8 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℝ) |
12 | 2re 10967 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ | |
13 | 12 | a1i 11 | . . . . . . . . 9 ⊢ (𝑃 ∈ ℕ → 2 ∈ ℝ) |
14 | 13, 9 | remulcld 9949 | . . . . . . . 8 ⊢ (𝑃 ∈ ℕ → (2 · 𝑃) ∈ ℝ) |
15 | 9 | ltm1d 10835 | . . . . . . . 8 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) < 𝑃) |
16 | nnnn0 11176 | . . . . . . . . . 10 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℕ0) | |
17 | 16 | nn0ge0d 11231 | . . . . . . . . 9 ⊢ (𝑃 ∈ ℕ → 0 ≤ 𝑃) |
18 | 1le2 11118 | . . . . . . . . . 10 ⊢ 1 ≤ 2 | |
19 | 18 | a1i 11 | . . . . . . . . 9 ⊢ (𝑃 ∈ ℕ → 1 ≤ 2) |
20 | 9, 13, 17, 19 | lemulge12d 10841 | . . . . . . . 8 ⊢ (𝑃 ∈ ℕ → 𝑃 ≤ (2 · 𝑃)) |
21 | 11, 9, 14, 15, 20 | ltletrd 10076 | . . . . . . 7 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) < (2 · 𝑃)) |
22 | 2pos 10989 | . . . . . . . . . 10 ⊢ 0 < 2 | |
23 | 12, 22 | pm3.2i 470 | . . . . . . . . 9 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
24 | 23 | a1i 11 | . . . . . . . 8 ⊢ (𝑃 ∈ ℕ → (2 ∈ ℝ ∧ 0 < 2)) |
25 | ltdivmul 10777 | . . . . . . . 8 ⊢ (((𝑃 − 1) ∈ ℝ ∧ 𝑃 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (((𝑃 − 1) / 2) < 𝑃 ↔ (𝑃 − 1) < (2 · 𝑃))) | |
26 | 11, 9, 24, 25 | syl3anc 1318 | . . . . . . 7 ⊢ (𝑃 ∈ ℕ → (((𝑃 − 1) / 2) < 𝑃 ↔ (𝑃 − 1) < (2 · 𝑃))) |
27 | 21, 26 | mpbird 246 | . . . . . 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 4618 | . . 3 ⊢ (𝜑 → 𝐻 < 𝑃) |
31 | prmndvdsfaclt 15273 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐻 ∈ ℕ0) → (𝐻 < 𝑃 → ¬ 𝑃 ∥ (!‘𝐻))) | |
32 | 7, 30, 31 | sylc 63 | . 2 ⊢ (𝜑 → ¬ 𝑃 ∥ (!‘𝐻)) |
33 | 6 | faccld 12933 | . . . . . 6 ⊢ (𝜑 → (!‘𝐻) ∈ ℕ) |
34 | 33 | nnzd 11357 | . . . . 5 ⊢ (𝜑 → (!‘𝐻) ∈ ℤ) |
35 | nnz 11276 | . . . . . . 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 15073 | . . . . 5 ⊢ (((!‘𝐻) ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((!‘𝐻) gcd 𝑃) = (𝑃 gcd (!‘𝐻))) | |
39 | 34, 37, 38 | syl2anc 691 | . . . 4 ⊢ (𝜑 → ((!‘𝐻) gcd 𝑃) = (𝑃 gcd (!‘𝐻))) |
40 | 39 | eqeq1d 2612 | . . 3 ⊢ (𝜑 → (((!‘𝐻) gcd 𝑃) = 1 ↔ (𝑃 gcd (!‘𝐻)) = 1)) |
41 | coprm 15261 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (!‘𝐻) ∈ ℤ) → (¬ 𝑃 ∥ (!‘𝐻) ↔ (𝑃 gcd (!‘𝐻)) = 1)) | |
42 | 3, 34, 41 | syl2anc 691 | . . 3 ⊢ (𝜑 → (¬ 𝑃 ∥ (!‘𝐻) ↔ (𝑃 gcd (!‘𝐻)) = 1)) |
43 | 40, 42 | bitr4d 270 | . 2 ⊢ (𝜑 → (((!‘𝐻) gcd 𝑃) = 1 ↔ ¬ 𝑃 ∥ (!‘𝐻))) |
44 | 32, 43 | mpbird 246 | 1 ⊢ (𝜑 → ((!‘𝐻) gcd 𝑃) = 1) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 {csn 4125 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 · cmul 9820 < clt 9953 ≤ cle 9954 − cmin 10145 / cdiv 10563 ℕcn 10897 2c2 10947 ℕ0cn0 11169 ℤcz 11254 !cfa 12922 ∥ cdvds 14821 gcd cgcd 15054 ℙcprime 15223 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-gcd 15055 df-prm 15224 |
This theorem is referenced by: gausslemma2dlem7 24898 |
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