Theorem gcdaddmlem 13734

Description: Lemma for gcdaddm 13735.

Hypotheses

Ref	Expression
gcdaddmlem.1	|- K e. ZZ
gcdaddmlem.2	|- M e. ZZ
gcdaddmlem.3	|- N e. ZZ

Assertion

Ref	Expression
gcdaddmlem	|- (M gcd N) = (M gcd ((K x. M) + N))

Proof of Theorem gcdaddmlem

Step	Hyp	Ref	Expression
1		gcdaddmlem.2	. . . . . . . 8 |- M e. ZZ
2		gcdaddmlem.3	. . . . . . . 8 |- N e. ZZ
3		gcddvds 13722	. . . . . . . 8 |- ((M e. ZZ /\ N e. ZZ) -> ((M gcd N)||M /\ (M gcd N)||N))
4	1, 2, 3	mp2an 761	. . . . . . 7 |- ((M gcd N)||M /\ (M gcd N)||N)
5	4	simpli 347	. . . . . 6 |- (M gcd N)||M
6		gcdcl 13724	. . . . . . . . . . 11 |- ((M e. ZZ /\ N e. ZZ) -> (M gcd N) e. NN0)
7	1, 2, 6	mp2an 761	. . . . . . . . . 10 |- (M gcd N) e. NN0
8		nn0z 7363	. . . . . . . . . 10 |- ((M gcd N) e. NN0 -> (M gcd N) e. ZZ)
9	7, 8	ax-mp 7	. . . . . . . . 9 |- (M gcd N) e. ZZ
10		gcdaddmlem.1	. . . . . . . . . 10 |- K e. ZZ
11		1z 7368	. . . . . . . . . 10 |- 1 e. ZZ
12		dvds2ln 13684	. . . . . . . . . 10 |- (((K e. ZZ /\ 1 e. ZZ) /\ ((M gcd N) e. ZZ /\ M e. ZZ /\ N e. ZZ)) -> (((M gcd N)||M /\ (M gcd N)||N) -> (M gcd N)||((K x. M) + (1 x. N))))
13	10, 11, 12	mpanl12 773	. . . . . . . . 9 |- (((M gcd N) e. ZZ /\ M e. ZZ /\ N e. ZZ) -> (((M gcd N)||M /\ (M gcd N)||N) -> (M gcd N)||((K x. M) + (1 x. N))))
14	9, 1, 2, 13	mp3an 1191	. . . . . . . 8 |- (((M gcd N)||M /\ (M gcd N)||N) -> (M gcd N)||((K x. M) + (1 x. N)))
15	4, 14	ax-mp 7	. . . . . . 7 |- (M gcd N)||((K x. M) + (1 x. N))
16		zcn 7349	. . . . . . . . . 10 |- (N e. ZZ -> N e. CC)
17	2, 16	ax-mp 7	. . . . . . . . 9 |- N e. CC
18	17	mulid2i 6486	. . . . . . . 8 |- (1 x. N) = N
19	18	opreq2i 4893	. . . . . . 7 |- ((K x. M) + (1 x. N)) = ((K x. M) + N)
20	15, 19	breqtri 3360	. . . . . 6 |- (M gcd N)||((K x. M) + N)
21	5, 20	pm3.2i 307	. . . . 5 |- ((M gcd N)||M /\ (M gcd N)||((K x. M) + N))
22		zmulcl 7389	. . . . . . . 8 |- ((K e. ZZ /\ M e. ZZ) -> (K x. M) e. ZZ)
23	10, 1, 22	mp2an 761	. . . . . . 7 |- (K x. M) e. ZZ
24		zaddcl 7374	. . . . . . 7 |- (((K x. M) e. ZZ /\ N e. ZZ) -> ((K x. M) + N) e. ZZ)
25	23, 2, 24	mp2an 761	. . . . . 6 |- ((K x. M) + N) e. ZZ
26		dvdslegcd 13723	. . . . . . 7 |- ((((M gcd N) e. ZZ /\ M e. ZZ /\ ((K x. M) + N) e. ZZ) /\ -. (M = 0 /\ ((K x. M) + N) = 0)) -> (((M gcd N)||M /\ (M gcd N)||((K x. M) + N)) -> (M gcd N) <_ (M gcd ((K x. M) + N))))
27	26	ex 402	. . . . . 6 |- (((M gcd N) e. ZZ /\ M e. ZZ /\ ((K x. M) + N) e. ZZ) -> (-. (M = 0 /\ ((K x. M) + N) = 0) -> (((M gcd N)||M /\ (M gcd N)||((K x. M) + N)) -> (M gcd N) <_ (M gcd ((K x. M) + N)))))
28	9, 1, 25, 27	mp3an 1191	. . . . 5 |- (-. (M = 0 /\ ((K x. M) + N) = 0) -> (((M gcd N)||M /\ (M gcd N)||((K x. M) + N)) -> (M gcd N) <_ (M gcd ((K x. M) + N))))
29	21, 28	mpi 55	. . . 4 |- (-. (M = 0 /\ ((K x. M) + N) = 0) -> (M gcd N) <_ (M gcd ((K x. M) + N)))
30		gcddvds 13722	. . . . . . . 8 |- ((M e. ZZ /\ ((K x. M) + N) e. ZZ) -> ((M gcd ((K x. M) + N))||M /\ (M gcd ((K x. M) + N))||((K x. M) + N)))
31	1, 25, 30	mp2an 761	. . . . . . 7 |- ((M gcd ((K x. M) + N))||M /\ (M gcd ((K x. M) + N))||((K x. M) + N))
32	31	simpli 347	. . . . . 6 |- (M gcd ((K x. M) + N))||M
33		gcdcl 13724	. . . . . . . . . . 11 |- ((M e. ZZ /\ ((K x. M) + N) e. ZZ) -> (M gcd ((K x. M) + N)) e. NN0)
34	1, 25, 33	mp2an 761	. . . . . . . . . 10 |- (M gcd ((K x. M) + N)) e. NN0
35		nn0z 7363	. . . . . . . . . 10 |- ((M gcd ((K x. M) + N)) e. NN0 -> (M gcd ((K x. M) + N)) e. ZZ)
36	34, 35	ax-mp 7	. . . . . . . . 9 |- (M gcd ((K x. M) + N)) e. ZZ
37		znegcl 7372	. . . . . . . . . . 11 |- (K e. ZZ -> -uK e. ZZ)
38	10, 37	ax-mp 7	. . . . . . . . . 10 |- -uK e. ZZ
39		dvds2ln 13684	. . . . . . . . . 10 |- (((-uK e. ZZ /\ 1 e. ZZ) /\ ((M gcd ((K x. M) + N)) e. ZZ /\ M e. ZZ /\ ((K x. M) + N) e. ZZ)) -> (((M gcd ((K x. M) + N))||M /\ (M gcd ((K x. M) + N))||((K x. M) + N)) -> (M gcd ((K x. M) + N))||((-uK x. M) + (1 x. ((K x. M) + N)))))
40	38, 11, 39	mpanl12 773	. . . . . . . . 9 |- (((M gcd ((K x. M) + N)) e. ZZ /\ M e. ZZ /\ ((K x. M) + N) e. ZZ) -> (((M gcd ((K x. M) + N))||M /\ (M gcd ((K x. M) + N))||((K x. M) + N)) -> (M gcd ((K x. M) + N))||((-uK x. M) + (1 x. ((K x. M) + N)))))
41	36, 1, 25, 40	mp3an 1191	. . . . . . . 8 |- (((M gcd ((K x. M) + N))||M /\ (M gcd ((K x. M) + N))||((K x. M) + N)) -> (M gcd ((K x. M) + N))||((-uK x. M) + (1 x. ((K x. M) + N))))
42	31, 41	ax-mp 7	. . . . . . 7 |- (M gcd ((K x. M) + N))||((-uK x. M) + (1 x. ((K x. M) + N)))
43		zcn 7349	. . . . . . . . . . 11 |- (K e. ZZ -> K e. CC)
44	10, 43	ax-mp 7	. . . . . . . . . 10 |- K e. CC
45		zcn 7349	. . . . . . . . . . 11 |- (M e. ZZ -> M e. CC)
46	1, 45	ax-mp 7	. . . . . . . . . 10 |- M e. CC
47	44, 46	mulneg1i 6608	. . . . . . . . 9 |- (-uK x. M) = -u(K x. M)
48		zcn 7349	. . . . . . . . . . 11 |- (((K x. M) + N) e. ZZ -> ((K x. M) + N) e. CC)
49	25, 48	ax-mp 7	. . . . . . . . . 10 |- ((K x. M) + N) e. CC
50	49	mulid2i 6486	. . . . . . . . 9 |- (1 x. ((K x. M) + N)) = ((K x. M) + N)
51	47, 50	opreq12i 4894	. . . . . . . 8 |- ((-uK x. M) + (1 x. ((K x. M) + N))) = (-u(K x. M) + ((K x. M) + N))
52	44, 46	mulcli 6474	. . . . . . . . . . . . 13 |- (K x. M) e. CC
53	52	negcli 6526	. . . . . . . . . . . 12 |- -u(K x. M) e. CC
54	53, 52	addcomi 6475	. . . . . . . . . . 11 |- (-u(K x. M) + (K x. M)) = ((K x. M) + -u(K x. M))
55	52	negidi 6537	. . . . . . . . . . 11 |- ((K x. M) + -u(K x. M)) = 0
56	54, 55	eqtri 1908	. . . . . . . . . 10 |- (-u(K x. M) + (K x. M)) = 0
57	56	opreq1i 4892	. . . . . . . . 9 |- ((-u(K x. M) + (K x. M)) + N) = (0 + N)
58	53, 52, 17	addassi 6477	. . . . . . . . 9 |- ((-u(K x. M) + (K x. M)) + N) = (-u(K x. M) + ((K x. M) + N))
59	17	addid2i 6484	. . . . . . . . 9 |- (0 + N) = N
60	57, 58, 59	3eqtr3i 1918	. . . . . . . 8 |- (-u(K x. M) + ((K x. M) + N)) = N
61	51, 60	eqtri 1908	. . . . . . 7 |- ((-uK x. M) + (1 x. ((K x. M) + N))) = N
62	42, 61	breqtri 3360	. . . . . 6 |- (M gcd ((K x. M) + N))||N
63	32, 62	pm3.2i 307	. . . . 5 |- ((M gcd ((K x. M) + N))||M /\ (M gcd ((K x. M) + N))||N)
64		dvdslegcd 13723	. . . . . . 7 |- ((((M gcd ((K x. M) + N)) e. ZZ /\ M e. ZZ /\ N e. ZZ) /\ -. (M = 0 /\ N = 0)) -> (((M gcd ((K x. M) + N))||M /\ (M gcd ((K x. M) + N))||N) -> (M gcd ((K x. M) + N)) <_ (M gcd N)))
65	64	ex 402	. . . . . 6 |- (((M gcd ((K x. M) + N)) e. ZZ /\ M e. ZZ /\ N e. ZZ) -> (-. (M = 0 /\ N = 0) -> (((M gcd ((K x. M) + N))||M /\ (M gcd ((K x. M) + N))||N) -> (M gcd ((K x. M) + N)) <_ (M gcd N))))
66	36, 1, 2, 65	mp3an 1191	. . . . 5 |- (-. (M = 0 /\ N = 0) -> (((M gcd ((K x. M) + N))||M /\ (M gcd ((K x. M) + N))||N) -> (M gcd ((K x. M) + N)) <_ (M gcd N)))
67	63, 66	mpi 55	. . . 4 |- (-. (M = 0 /\ N = 0) -> (M gcd ((K x. M) + N)) <_ (M gcd N))
68	29, 67	anim12i 360	. . 3 |- ((-. (M = 0 /\ ((K x. M) + N) = 0) /\ -. (M = 0 /\ N = 0)) -> ((M gcd N) <_ (M gcd ((K x. M) + N)) /\ (M gcd ((K x. M) + N)) <_ (M gcd N)))
69		zre 7348	. . . . 5 |- ((M gcd N) e. ZZ -> (M gcd N) e. RR)
70	9, 69	ax-mp 7	. . . 4 |- (M gcd N) e. RR
71		zre 7348	. . . . 5 |- ((M gcd ((K x. M) + N)) e. ZZ -> (M gcd ((K x. M) + N)) e. RR)
72	36, 71	ax-mp 7	. . . 4 |- (M gcd ((K x. M) + N)) e. RR
73	70, 72	letri3i 6749	. . 3 |- ((M gcd N) = (M gcd ((K x. M) + N)) <-> ((M gcd N) <_ (M gcd ((K x. M) + N)) /\ (M gcd ((K x. M) + N)) <_ (M gcd N)))
74	68, 73	sylibr 217	. 2 |- ((-. (M = 0 /\ ((K x. M) + N) = 0) /\ -. (M = 0 /\ N = 0)) -> (M gcd N) = (M gcd ((K x. M) + N)))
75		pm4.57 340	. . 3 |- (-. (-. (M = 0 /\ ((K x. M) + N) = 0) /\ -. (M = 0 /\ N = 0)) <-> ((M = 0 /\ ((K x. M) + N) = 0) \/ (M = 0 /\ N = 0)))
76		opreq2 4890	. . . . . . . . . 10 |- (M = 0 -> (K x. M) = (K x. 0))
77	44	mul01i 6594	. . . . . . . . . 10 |- (K x. 0) = 0
78	76, 77	syl6eq 1944	. . . . . . . . 9 |- (M = 0 -> (K x. M) = 0)
79	78	opreq1d 4897	. . . . . . . 8 |- (M = 0 -> ((K x. M) + N) = (0 + N))
80	79, 59	syl6eq 1944	. . . . . . 7 |- (M = 0 -> ((K x. M) + N) = N)
81	80	eqeq1d 1892	. . . . . 6 |- (M = 0 -> (((K x. M) + N) = 0 <-> N = 0))
82	81	pm5.32i 707	. . . . 5 |- ((M = 0 /\ ((K x. M) + N) = 0) <-> (M = 0 /\ N = 0))
83		opreq12 4891	. . . . . 6 |- ((M = 0 /\ N = 0) -> (M gcd N) = (0 gcd 0))
84		opreq12 4891	. . . . . . 7 |- ((M = 0 /\ ((K x. M) + N) = 0) -> (M gcd ((K x. M) + N)) = (0 gcd 0))
85	82, 84	sylbir 218	. . . . . 6 |- ((M = 0 /\ N = 0) -> (M gcd ((K x. M) + N)) = (0 gcd 0))
86	83, 85	eqtr4d 1928	. . . . 5 |- ((M = 0 /\ N = 0) -> (M gcd N) = (M gcd ((K x. M) + N)))
87	82, 86	sylbi 216	. . . 4 |- ((M = 0 /\ ((K x. M) + N) = 0) -> (M gcd N) = (M gcd ((K x. M) + N)))
88	87, 86	jaoi 368	. . 3 |- (((M = 0 /\ ((K x. M) + N) = 0) \/ (M = 0 /\ N = 0)) -> (M gcd N) = (M gcd ((K x. M) + N)))
89	75, 88	sylbi 216	. 2 |- (-. (-. (M = 0 /\ ((K x. M) + N) = 0) /\ -. (M = 0 /\ N = 0)) -> (M gcd N) = (M gcd ((K x. M) + N)))
90	74, 89	pm2.61i 140	1 |- (M gcd N) = (M gcd ((K x. M) + N))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 239   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   class class class wbr 3338  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389   x. cmul 6391  -ucneg 6446   <_ cle 6448  NN0cn0 6450  ZZcz 6451  ||cdivides 13662   gcd cgcd 13713

This theorem is referenced by:  gcdaddm 13735

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-n0 7309  df-z 7345  df-uz 7587  df-seq1 7721  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-divides 13663  df-gcd 13714

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