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Mirrors > Home > MPE Home > Th. List > gausslemma2dlem0c | Structured version Visualization version GIF version |
Description: Auxiliary lemma 3 for gausslemma2d 27433. (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 4141 | . . . . 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 27416 | . . . . 5 ⊢ (𝜑 → 𝐻 ∈ ℕ) |
6 | 5 | nnnn0d 12585 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ ℕ0) |
7 | 3, 6 | jca 511 | . . 3 ⊢ (𝜑 → (𝑃 ∈ ℙ ∧ 𝐻 ∈ ℕ0)) |
8 | prmnn 16708 | . . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | |
9 | nnre 12271 | . . . . . . . 8 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℝ) | |
10 | peano2rem 11574 | . . . . . . . 8 ⊢ (𝑃 ∈ ℝ → (𝑃 − 1) ∈ ℝ) | |
11 | 9, 10 | syl 17 | . . . . . . 7 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℝ) |
12 | 2re 12338 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
13 | 12 | a1i 11 | . . . . . . . 8 ⊢ (𝑃 ∈ ℕ → 2 ∈ ℝ) |
14 | 13, 9 | remulcld 11289 | . . . . . . 7 ⊢ (𝑃 ∈ ℕ → (2 · 𝑃) ∈ ℝ) |
15 | 9 | ltm1d 12198 | . . . . . . 7 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) < 𝑃) |
16 | nnnn0 12531 | . . . . . . . . 9 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℕ0) | |
17 | 16 | nn0ge0d 12588 | . . . . . . . 8 ⊢ (𝑃 ∈ ℕ → 0 ≤ 𝑃) |
18 | 1le2 12473 | . . . . . . . . 9 ⊢ 1 ≤ 2 | |
19 | 18 | a1i 11 | . . . . . . . 8 ⊢ (𝑃 ∈ ℕ → 1 ≤ 2) |
20 | 9, 13, 17, 19 | lemulge12d 12204 | . . . . . . 7 ⊢ (𝑃 ∈ ℕ → 𝑃 ≤ (2 · 𝑃)) |
21 | 11, 9, 14, 15, 20 | ltletrd 11419 | . . . . . 6 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) < (2 · 𝑃)) |
22 | 2pos 12367 | . . . . . . . . 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 12141 | . . . . . . 7 ⊢ (((𝑃 − 1) ∈ ℝ ∧ 𝑃 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (((𝑃 − 1) / 2) < 𝑃 ↔ (𝑃 − 1) < (2 · 𝑃))) | |
26 | 11, 9, 24, 25 | syl3anc 1370 | . . . . . 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 5183 | . . 3 ⊢ (𝜑 → 𝐻 < 𝑃) |
30 | prmndvdsfaclt 16759 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐻 ∈ ℕ0) → (𝐻 < 𝑃 → ¬ 𝑃 ∥ (!‘𝐻))) | |
31 | 7, 29, 30 | sylc 65 | . 2 ⊢ (𝜑 → ¬ 𝑃 ∥ (!‘𝐻)) |
32 | 6 | faccld 14320 | . . . . . 6 ⊢ (𝜑 → (!‘𝐻) ∈ ℕ) |
33 | 32 | nnzd 12638 | . . . . 5 ⊢ (𝜑 → (!‘𝐻) ∈ ℤ) |
34 | nnz 12632 | . . . . . 6 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℤ) | |
35 | 1, 2, 8, 34 | 4syl 19 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℤ) |
36 | 33, 35 | gcdcomd 16548 | . . . 4 ⊢ (𝜑 → ((!‘𝐻) gcd 𝑃) = (𝑃 gcd (!‘𝐻))) |
37 | 36 | eqeq1d 2737 | . . 3 ⊢ (𝜑 → (((!‘𝐻) gcd 𝑃) = 1 ↔ (𝑃 gcd (!‘𝐻)) = 1)) |
38 | coprm 16745 | . . . 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 1537 ∈ wcel 2106 ∖ cdif 3960 {csn 4631 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 0cc0 11153 1c1 11154 · cmul 11158 < clt 11293 ≤ cle 11294 − cmin 11490 / cdiv 11918 ℕcn 12264 2c2 12319 ℕ0cn0 12524 ℤcz 12611 !cfa 14309 ∥ cdvds 16287 gcd cgcd 16528 ℙcprime 16705 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-dvds 16288 df-gcd 16529 df-prm 16706 |
This theorem is referenced by: gausslemma2dlem7 27432 |
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