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Mirrors > Home > MPE Home > Th. List > prmgapprmolem | Structured version Visualization version GIF version |
Description: Lemma for prmgapprmo 17025: The primorial of a number plus an integer greater than 1 and less then or equal to the number are not coprime. (Contributed by AV, 15-Aug-2020.) (Revised by AV, 29-Aug-2020.) |
Ref | Expression |
---|---|
prmgapprmolem | ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 1 < (((#p‘𝑁) + 𝐼) gcd 𝐼)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prmuz2 16661 | . . . . 5 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ≥‘2)) | |
2 | 1 | ad2antlr 725 | . . . 4 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) ∧ 𝑝 ∈ ℙ) ∧ (𝑝 ≤ 𝑁 ∧ 𝑝 ∥ 𝐼 ∧ 𝑝 ∥ ((#p‘𝑁) + 𝐼))) → 𝑝 ∈ (ℤ≥‘2)) |
3 | breq1 5147 | . . . . . 6 ⊢ (𝑞 = 𝑝 → (𝑞 ∥ ((#p‘𝑁) + 𝐼) ↔ 𝑝 ∥ ((#p‘𝑁) + 𝐼))) | |
4 | breq1 5147 | . . . . . 6 ⊢ (𝑞 = 𝑝 → (𝑞 ∥ 𝐼 ↔ 𝑝 ∥ 𝐼)) | |
5 | 3, 4 | anbi12d 630 | . . . . 5 ⊢ (𝑞 = 𝑝 → ((𝑞 ∥ ((#p‘𝑁) + 𝐼) ∧ 𝑞 ∥ 𝐼) ↔ (𝑝 ∥ ((#p‘𝑁) + 𝐼) ∧ 𝑝 ∥ 𝐼))) |
6 | 5 | adantl 480 | . . . 4 ⊢ (((((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) ∧ 𝑝 ∈ ℙ) ∧ (𝑝 ≤ 𝑁 ∧ 𝑝 ∥ 𝐼 ∧ 𝑝 ∥ ((#p‘𝑁) + 𝐼))) ∧ 𝑞 = 𝑝) → ((𝑞 ∥ ((#p‘𝑁) + 𝐼) ∧ 𝑞 ∥ 𝐼) ↔ (𝑝 ∥ ((#p‘𝑁) + 𝐼) ∧ 𝑝 ∥ 𝐼))) |
7 | pm3.22 458 | . . . . . 6 ⊢ ((𝑝 ∥ 𝐼 ∧ 𝑝 ∥ ((#p‘𝑁) + 𝐼)) → (𝑝 ∥ ((#p‘𝑁) + 𝐼) ∧ 𝑝 ∥ 𝐼)) | |
8 | 7 | 3adant1 1127 | . . . . 5 ⊢ ((𝑝 ≤ 𝑁 ∧ 𝑝 ∥ 𝐼 ∧ 𝑝 ∥ ((#p‘𝑁) + 𝐼)) → (𝑝 ∥ ((#p‘𝑁) + 𝐼) ∧ 𝑝 ∥ 𝐼)) |
9 | 8 | adantl 480 | . . . 4 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) ∧ 𝑝 ∈ ℙ) ∧ (𝑝 ≤ 𝑁 ∧ 𝑝 ∥ 𝐼 ∧ 𝑝 ∥ ((#p‘𝑁) + 𝐼))) → (𝑝 ∥ ((#p‘𝑁) + 𝐼) ∧ 𝑝 ∥ 𝐼)) |
10 | 2, 6, 9 | rspcedvd 3605 | . . 3 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) ∧ 𝑝 ∈ ℙ) ∧ (𝑝 ≤ 𝑁 ∧ 𝑝 ∥ 𝐼 ∧ 𝑝 ∥ ((#p‘𝑁) + 𝐼))) → ∃𝑞 ∈ (ℤ≥‘2)(𝑞 ∥ ((#p‘𝑁) + 𝐼) ∧ 𝑞 ∥ 𝐼)) |
11 | prmdvdsprmop 17006 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → ∃𝑝 ∈ ℙ (𝑝 ≤ 𝑁 ∧ 𝑝 ∥ 𝐼 ∧ 𝑝 ∥ ((#p‘𝑁) + 𝐼))) | |
12 | 10, 11 | r19.29a 3152 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → ∃𝑞 ∈ (ℤ≥‘2)(𝑞 ∥ ((#p‘𝑁) + 𝐼) ∧ 𝑞 ∥ 𝐼)) |
13 | nnnn0 12504 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
14 | prmocl 16997 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (#p‘𝑁) ∈ ℕ) | |
15 | 13, 14 | syl 17 | . . . 4 ⊢ (𝑁 ∈ ℕ → (#p‘𝑁) ∈ ℕ) |
16 | elfzuz 13524 | . . . . 5 ⊢ (𝐼 ∈ (2...𝑁) → 𝐼 ∈ (ℤ≥‘2)) | |
17 | eluz2nn 12893 | . . . . 5 ⊢ (𝐼 ∈ (ℤ≥‘2) → 𝐼 ∈ ℕ) | |
18 | 16, 17 | syl 17 | . . . 4 ⊢ (𝐼 ∈ (2...𝑁) → 𝐼 ∈ ℕ) |
19 | nnaddcl 12260 | . . . 4 ⊢ (((#p‘𝑁) ∈ ℕ ∧ 𝐼 ∈ ℕ) → ((#p‘𝑁) + 𝐼) ∈ ℕ) | |
20 | 15, 18, 19 | syl2an 594 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → ((#p‘𝑁) + 𝐼) ∈ ℕ) |
21 | 18 | adantl 480 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 𝐼 ∈ ℕ) |
22 | ncoprmgcdgt1b 16616 | . . 3 ⊢ ((((#p‘𝑁) + 𝐼) ∈ ℕ ∧ 𝐼 ∈ ℕ) → (∃𝑞 ∈ (ℤ≥‘2)(𝑞 ∥ ((#p‘𝑁) + 𝐼) ∧ 𝑞 ∥ 𝐼) ↔ 1 < (((#p‘𝑁) + 𝐼) gcd 𝐼))) | |
23 | 20, 21, 22 | syl2anc 582 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → (∃𝑞 ∈ (ℤ≥‘2)(𝑞 ∥ ((#p‘𝑁) + 𝐼) ∧ 𝑞 ∥ 𝐼) ↔ 1 < (((#p‘𝑁) + 𝐼) gcd 𝐼))) |
24 | 12, 23 | mpbid 231 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 1 < (((#p‘𝑁) + 𝐼) gcd 𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2098 ∃wrex 3060 class class class wbr 5144 ‘cfv 6543 (class class class)co 7413 1c1 11134 + caddc 11136 < clt 11273 ≤ cle 11274 ℕcn 12237 2c2 12292 ℕ0cn0 12497 ℤ≥cuz 12847 ...cfz 13511 ∥ cdvds 16225 gcd cgcd 16463 ℙcprime 16636 #pcprmo 16994 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9460 df-inf 9461 df-oi 9528 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-n0 12498 df-z 12584 df-uz 12848 df-rp 13002 df-fz 13512 df-fzo 13655 df-seq 13994 df-exp 14054 df-hash 14317 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-clim 15459 df-prod 15877 df-dvds 16226 df-gcd 16464 df-prm 16637 df-prmo 16995 |
This theorem is referenced by: prmgapprmo 17025 |
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