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Mirrors > Home > MPE Home > Th. List > Mathboxes > jm2.19lem2 | Structured version Visualization version GIF version |
Description: Lemma for jm2.19 36578. (Contributed by Stefan O'Rear, 23-Sep-2014.) |
Ref | Expression |
---|---|
jm2.19lem2 | ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐴 Yrm 𝑀) ∥ (𝐴 Yrm 𝑁) ↔ (𝐴 Yrm 𝑀) ∥ (𝐴 Yrm (𝑁 + 𝑀)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frmy 36497 | . . . . . 6 ⊢ Yrm :((ℤ≥‘2) × ℤ)⟶ℤ | |
2 | 1 | fovcl 6663 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → (𝐴 Yrm 𝑀) ∈ ℤ) |
3 | 2 | 3adant3 1074 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm 𝑀) ∈ ℤ) |
4 | 1 | fovcl 6663 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm 𝑁) ∈ ℤ) |
5 | 4 | 3adant2 1073 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm 𝑁) ∈ ℤ) |
6 | frmx 36496 | . . . . . . 7 ⊢ Xrm :((ℤ≥‘2) × ℤ)⟶ℕ0 | |
7 | 6 | fovcl 6663 | . . . . . 6 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → (𝐴 Xrm 𝑀) ∈ ℕ0) |
8 | 7 | 3adant3 1074 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm 𝑀) ∈ ℕ0) |
9 | 8 | nn0zd 11356 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm 𝑀) ∈ ℤ) |
10 | gcdcom 15073 | . . . . . 6 ⊢ (((𝐴 Yrm 𝑀) ∈ ℤ ∧ (𝐴 Xrm 𝑀) ∈ ℤ) → ((𝐴 Yrm 𝑀) gcd (𝐴 Xrm 𝑀)) = ((𝐴 Xrm 𝑀) gcd (𝐴 Yrm 𝑀))) | |
11 | 3, 9, 10 | syl2anc 691 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐴 Yrm 𝑀) gcd (𝐴 Xrm 𝑀)) = ((𝐴 Xrm 𝑀) gcd (𝐴 Yrm 𝑀))) |
12 | jm2.19lem1 36574 | . . . . . 6 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → ((𝐴 Xrm 𝑀) gcd (𝐴 Yrm 𝑀)) = 1) | |
13 | 12 | 3adant3 1074 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐴 Xrm 𝑀) gcd (𝐴 Yrm 𝑀)) = 1) |
14 | 11, 13 | eqtrd 2644 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐴 Yrm 𝑀) gcd (𝐴 Xrm 𝑀)) = 1) |
15 | coprmdvdsb 36570 | . . . 4 ⊢ (((𝐴 Yrm 𝑀) ∈ ℤ ∧ (𝐴 Yrm 𝑁) ∈ ℤ ∧ ((𝐴 Xrm 𝑀) ∈ ℤ ∧ ((𝐴 Yrm 𝑀) gcd (𝐴 Xrm 𝑀)) = 1)) → ((𝐴 Yrm 𝑀) ∥ (𝐴 Yrm 𝑁) ↔ (𝐴 Yrm 𝑀) ∥ ((𝐴 Xrm 𝑀) · (𝐴 Yrm 𝑁)))) | |
16 | 3, 5, 9, 14, 15 | syl112anc 1322 | . . 3 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐴 Yrm 𝑀) ∥ (𝐴 Yrm 𝑁) ↔ (𝐴 Yrm 𝑀) ∥ ((𝐴 Xrm 𝑀) · (𝐴 Yrm 𝑁)))) |
17 | 8 | nn0cnd 11230 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm 𝑀) ∈ ℂ) |
18 | 5 | zcnd 11359 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm 𝑁) ∈ ℂ) |
19 | 17, 18 | mulcomd 9940 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐴 Xrm 𝑀) · (𝐴 Yrm 𝑁)) = ((𝐴 Yrm 𝑁) · (𝐴 Xrm 𝑀))) |
20 | 19 | breq2d 4595 | . . 3 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐴 Yrm 𝑀) ∥ ((𝐴 Xrm 𝑀) · (𝐴 Yrm 𝑁)) ↔ (𝐴 Yrm 𝑀) ∥ ((𝐴 Yrm 𝑁) · (𝐴 Xrm 𝑀)))) |
21 | 16, 20 | bitrd 267 | . 2 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐴 Yrm 𝑀) ∥ (𝐴 Yrm 𝑁) ↔ (𝐴 Yrm 𝑀) ∥ ((𝐴 Yrm 𝑁) · (𝐴 Xrm 𝑀)))) |
22 | 5, 9 | zmulcld 11364 | . . 3 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐴 Yrm 𝑁) · (𝐴 Xrm 𝑀)) ∈ ℤ) |
23 | 6 | fovcl 6663 | . . . . . 6 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm 𝑁) ∈ ℕ0) |
24 | 23 | 3adant2 1073 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm 𝑁) ∈ ℕ0) |
25 | 24 | nn0zd 11356 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm 𝑁) ∈ ℤ) |
26 | 25, 3 | zmulcld 11364 | . . 3 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐴 Xrm 𝑁) · (𝐴 Yrm 𝑀)) ∈ ℤ) |
27 | dvdsmul2 14842 | . . . 4 ⊢ (((𝐴 Xrm 𝑁) ∈ ℤ ∧ (𝐴 Yrm 𝑀) ∈ ℤ) → (𝐴 Yrm 𝑀) ∥ ((𝐴 Xrm 𝑁) · (𝐴 Yrm 𝑀))) | |
28 | 25, 3, 27 | syl2anc 691 | . . 3 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm 𝑀) ∥ ((𝐴 Xrm 𝑁) · (𝐴 Yrm 𝑀))) |
29 | dvdsadd2b 14866 | . . 3 ⊢ (((𝐴 Yrm 𝑀) ∈ ℤ ∧ ((𝐴 Yrm 𝑁) · (𝐴 Xrm 𝑀)) ∈ ℤ ∧ (((𝐴 Xrm 𝑁) · (𝐴 Yrm 𝑀)) ∈ ℤ ∧ (𝐴 Yrm 𝑀) ∥ ((𝐴 Xrm 𝑁) · (𝐴 Yrm 𝑀)))) → ((𝐴 Yrm 𝑀) ∥ ((𝐴 Yrm 𝑁) · (𝐴 Xrm 𝑀)) ↔ (𝐴 Yrm 𝑀) ∥ (((𝐴 Xrm 𝑁) · (𝐴 Yrm 𝑀)) + ((𝐴 Yrm 𝑁) · (𝐴 Xrm 𝑀))))) | |
30 | 3, 22, 26, 28, 29 | syl112anc 1322 | . 2 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐴 Yrm 𝑀) ∥ ((𝐴 Yrm 𝑁) · (𝐴 Xrm 𝑀)) ↔ (𝐴 Yrm 𝑀) ∥ (((𝐴 Xrm 𝑁) · (𝐴 Yrm 𝑀)) + ((𝐴 Yrm 𝑁) · (𝐴 Xrm 𝑀))))) |
31 | rmyadd 36514 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐴 Yrm (𝑁 + 𝑀)) = (((𝐴 Yrm 𝑁) · (𝐴 Xrm 𝑀)) + ((𝐴 Xrm 𝑁) · (𝐴 Yrm 𝑀)))) | |
32 | 31 | 3com23 1263 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm (𝑁 + 𝑀)) = (((𝐴 Yrm 𝑁) · (𝐴 Xrm 𝑀)) + ((𝐴 Xrm 𝑁) · (𝐴 Yrm 𝑀)))) |
33 | 18, 17 | mulcld 9939 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐴 Yrm 𝑁) · (𝐴 Xrm 𝑀)) ∈ ℂ) |
34 | 24 | nn0cnd 11230 | . . . . . 6 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm 𝑁) ∈ ℂ) |
35 | 3 | zcnd 11359 | . . . . . 6 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm 𝑀) ∈ ℂ) |
36 | 34, 35 | mulcld 9939 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐴 Xrm 𝑁) · (𝐴 Yrm 𝑀)) ∈ ℂ) |
37 | 33, 36 | addcomd 10117 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝐴 Yrm 𝑁) · (𝐴 Xrm 𝑀)) + ((𝐴 Xrm 𝑁) · (𝐴 Yrm 𝑀))) = (((𝐴 Xrm 𝑁) · (𝐴 Yrm 𝑀)) + ((𝐴 Yrm 𝑁) · (𝐴 Xrm 𝑀)))) |
38 | 32, 37 | eqtr2d 2645 | . . 3 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝐴 Xrm 𝑁) · (𝐴 Yrm 𝑀)) + ((𝐴 Yrm 𝑁) · (𝐴 Xrm 𝑀))) = (𝐴 Yrm (𝑁 + 𝑀))) |
39 | 38 | breq2d 4595 | . 2 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐴 Yrm 𝑀) ∥ (((𝐴 Xrm 𝑁) · (𝐴 Yrm 𝑀)) + ((𝐴 Yrm 𝑁) · (𝐴 Xrm 𝑀))) ↔ (𝐴 Yrm 𝑀) ∥ (𝐴 Yrm (𝑁 + 𝑀)))) |
40 | 21, 30, 39 | 3bitrd 293 | 1 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐴 Yrm 𝑀) ∥ (𝐴 Yrm 𝑁) ↔ (𝐴 Yrm 𝑀) ∥ (𝐴 Yrm (𝑁 + 𝑀)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 1c1 9816 + caddc 9818 · cmul 9820 2c2 10947 ℕ0cn0 11169 ℤcz 11254 ℤ≥cuz 11563 ∥ cdvds 14821 gcd cgcd 15054 Xrm crmx 36482 Yrm crmy 36483 |
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-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 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 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 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-iin 4458 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-se 4998 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-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-cda 8873 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-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-xnn0 11241 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 df-pi 14642 df-dvds 14822 df-gcd 15055 df-numer 15281 df-denom 15282 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-limc 23436 df-dv 23437 df-log 24107 df-squarenn 36423 df-pell1qr 36424 df-pell14qr 36425 df-pell1234qr 36426 df-pellfund 36427 df-rmx 36484 df-rmy 36485 |
This theorem is referenced by: jm2.19lem3 36576 |
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