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Mirrors > Home > MPE Home > Th. List > Mathboxes > jm2.19lem1 | Structured version Visualization version GIF version |
Description: Lemma for jm2.19 42982. X and Y values are coprime. (Contributed by Stefan O'Rear, 23-Sep-2014.) |
Ref | Expression |
---|---|
jm2.19lem1 | ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → ((𝐴 Xrm 𝑀) gcd (𝐴 Yrm 𝑀)) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frmx 42902 | . . . . . . 7 ⊢ Xrm :((ℤ≥‘2) × ℤ)⟶ℕ0 | |
2 | 1 | fovcl 7561 | . . . . . 6 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → (𝐴 Xrm 𝑀) ∈ ℕ0) |
3 | 2 | nn0cnd 12587 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → (𝐴 Xrm 𝑀) ∈ ℂ) |
4 | 3 | sqcld 14181 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → ((𝐴 Xrm 𝑀)↑2) ∈ ℂ) |
5 | rmspecnonsq 42895 | . . . . . . . 8 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ (ℕ ∖ ◻NN)) | |
6 | 5 | eldifad 3975 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ ℕ) |
7 | 6 | adantr 480 | . . . . . 6 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → ((𝐴↑2) − 1) ∈ ℕ) |
8 | 7 | nncnd 12280 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → ((𝐴↑2) − 1) ∈ ℂ) |
9 | frmy 42903 | . . . . . . . 8 ⊢ Yrm :((ℤ≥‘2) × ℤ)⟶ℤ | |
10 | 9 | fovcl 7561 | . . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → (𝐴 Yrm 𝑀) ∈ ℤ) |
11 | 10 | zcnd 12721 | . . . . . 6 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → (𝐴 Yrm 𝑀) ∈ ℂ) |
12 | 11 | sqcld 14181 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → ((𝐴 Yrm 𝑀)↑2) ∈ ℂ) |
13 | 8, 12 | mulcld 11279 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → (((𝐴↑2) − 1) · ((𝐴 Yrm 𝑀)↑2)) ∈ ℂ) |
14 | 4, 13 | negsubd 11624 | . . 3 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → (((𝐴 Xrm 𝑀)↑2) + -(((𝐴↑2) − 1) · ((𝐴 Yrm 𝑀)↑2))) = (((𝐴 Xrm 𝑀)↑2) − (((𝐴↑2) − 1) · ((𝐴 Yrm 𝑀)↑2)))) |
15 | 3 | sqvald 14180 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → ((𝐴 Xrm 𝑀)↑2) = ((𝐴 Xrm 𝑀) · (𝐴 Xrm 𝑀))) |
16 | 11 | sqvald 14180 | . . . . . 6 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → ((𝐴 Yrm 𝑀)↑2) = ((𝐴 Yrm 𝑀) · (𝐴 Yrm 𝑀))) |
17 | 16 | oveq2d 7447 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → (-((𝐴↑2) − 1) · ((𝐴 Yrm 𝑀)↑2)) = (-((𝐴↑2) − 1) · ((𝐴 Yrm 𝑀) · (𝐴 Yrm 𝑀)))) |
18 | 8, 12 | mulneg1d 11714 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → (-((𝐴↑2) − 1) · ((𝐴 Yrm 𝑀)↑2)) = -(((𝐴↑2) − 1) · ((𝐴 Yrm 𝑀)↑2))) |
19 | nnnegz 12614 | . . . . . . . 8 ⊢ (((𝐴↑2) − 1) ∈ ℕ → -((𝐴↑2) − 1) ∈ ℤ) | |
20 | 7, 19 | syl 17 | . . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → -((𝐴↑2) − 1) ∈ ℤ) |
21 | 20 | zcnd 12721 | . . . . . 6 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → -((𝐴↑2) − 1) ∈ ℂ) |
22 | 21, 11, 11 | mul12d 11468 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → (-((𝐴↑2) − 1) · ((𝐴 Yrm 𝑀) · (𝐴 Yrm 𝑀))) = ((𝐴 Yrm 𝑀) · (-((𝐴↑2) − 1) · (𝐴 Yrm 𝑀)))) |
23 | 17, 18, 22 | 3eqtr3d 2783 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → -(((𝐴↑2) − 1) · ((𝐴 Yrm 𝑀)↑2)) = ((𝐴 Yrm 𝑀) · (-((𝐴↑2) − 1) · (𝐴 Yrm 𝑀)))) |
24 | 15, 23 | oveq12d 7449 | . . 3 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → (((𝐴 Xrm 𝑀)↑2) + -(((𝐴↑2) − 1) · ((𝐴 Yrm 𝑀)↑2))) = (((𝐴 Xrm 𝑀) · (𝐴 Xrm 𝑀)) + ((𝐴 Yrm 𝑀) · (-((𝐴↑2) − 1) · (𝐴 Yrm 𝑀))))) |
25 | rmxynorm 42907 | . . 3 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → (((𝐴 Xrm 𝑀)↑2) − (((𝐴↑2) − 1) · ((𝐴 Yrm 𝑀)↑2))) = 1) | |
26 | 14, 24, 25 | 3eqtr3d 2783 | . 2 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → (((𝐴 Xrm 𝑀) · (𝐴 Xrm 𝑀)) + ((𝐴 Yrm 𝑀) · (-((𝐴↑2) − 1) · (𝐴 Yrm 𝑀)))) = 1) |
27 | 2 | nn0zd 12637 | . . 3 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → (𝐴 Xrm 𝑀) ∈ ℤ) |
28 | 20, 10 | zmulcld 12726 | . . 3 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → (-((𝐴↑2) − 1) · (𝐴 Yrm 𝑀)) ∈ ℤ) |
29 | bezoutr1 16603 | . . 3 ⊢ ((((𝐴 Xrm 𝑀) ∈ ℤ ∧ (𝐴 Yrm 𝑀) ∈ ℤ) ∧ ((𝐴 Xrm 𝑀) ∈ ℤ ∧ (-((𝐴↑2) − 1) · (𝐴 Yrm 𝑀)) ∈ ℤ)) → ((((𝐴 Xrm 𝑀) · (𝐴 Xrm 𝑀)) + ((𝐴 Yrm 𝑀) · (-((𝐴↑2) − 1) · (𝐴 Yrm 𝑀)))) = 1 → ((𝐴 Xrm 𝑀) gcd (𝐴 Yrm 𝑀)) = 1)) | |
30 | 27, 10, 27, 28, 29 | syl22anc 839 | . 2 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → ((((𝐴 Xrm 𝑀) · (𝐴 Xrm 𝑀)) + ((𝐴 Yrm 𝑀) · (-((𝐴↑2) − 1) · (𝐴 Yrm 𝑀)))) = 1 → ((𝐴 Xrm 𝑀) gcd (𝐴 Yrm 𝑀)) = 1)) |
31 | 26, 30 | mpd 15 | 1 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → ((𝐴 Xrm 𝑀) gcd (𝐴 Yrm 𝑀)) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 1c1 11154 + caddc 11156 · cmul 11158 − cmin 11490 -cneg 11491 ℕcn 12264 2c2 12319 ℕ0cn0 12524 ℤcz 12611 ℤ≥cuz 12876 ↑cexp 14099 gcd cgcd 16528 ◻NNcsquarenn 42824 Xrm crmx 42888 Yrm crmy 42889 |
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-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 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 ax-addf 11232 |
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-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 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-se 5642 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-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-omul 8510 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-acn 9980 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-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12598 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-sin 16102 df-cos 16103 df-pi 16105 df-dvds 16288 df-gcd 16529 df-numer 16769 df-denom 16770 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-limc 25916 df-dv 25917 df-log 26613 df-squarenn 42829 df-pell1qr 42830 df-pell14qr 42831 df-pell1234qr 42832 df-pellfund 42833 df-rmx 42890 df-rmy 42891 |
This theorem is referenced by: jm2.19lem2 42979 jm2.20nn 42986 |
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