Theorem mulgcdlem7 13762

Description: Lemma for mulgcd 13763.

Assertion

Ref	Expression
mulgcdlem7	|- ((K e. NN0 /\ M e. NN0 /\ N e. NN0) -> ((K x. M) gcd (K x. N)) = (K x. (M gcd N)))

Proof of Theorem mulgcdlem7

Step	Hyp	Ref	Expression
1		nn0ge0 7326	. . . . 5 |- (K e. NN0 -> 0 <_ K)
2		nn0re 7317	. . . . . 6 |- (K e. NN0 -> K e. RR)
3		0re 6603	. . . . . . 7 |- 0 e. RR
4		leloe 6688	. . . . . . 7 |- ((0 e. RR /\ K e. RR) -> (0 <_ K <-> (0 < K \/ 0 = K)))
5	3, 4	mpan 759	. . . . . 6 |- (K e. RR -> (0 <_ K <-> (0 < K \/ 0 = K)))
6	2, 5	syl 12	. . . . 5 |- (K e. NN0 -> (0 <_ K <-> (0 < K \/ 0 = K)))
7	1, 6	mpbid 212	. . . 4 |- (K e. NN0 -> (0 < K \/ 0 = K))
8	7	adantr 425	. . 3 |- ((K e. NN0 /\ (M e. NN0 /\ N e. NN0)) -> (0 < K \/ 0 = K))
9		mulgcdlem6 13761	. . . . . . . 8 |- ((K e. NN /\ M e. NN0 /\ N e. NN0) -> ((K x. M) gcd (K x. N)) = (K x. (M gcd N)))
10	9	3expb 1068	. . . . . . 7 |- ((K e. NN /\ (M e. NN0 /\ N e. NN0)) -> ((K x. M) gcd (K x. N)) = (K x. (M gcd N)))
11		elnnz 7354	. . . . . . . . . 10 |- (K e. NN <-> (K e. ZZ /\ 0 < K))
12	11	biimpri 169	. . . . . . . . 9 |- ((K e. ZZ /\ 0 < K) -> K e. NN)
13		nn0z 7363	. . . . . . . . 9 |- (K e. NN0 -> K e. ZZ)
14	12, 13	sylan 497	. . . . . . . 8 |- ((K e. NN0 /\ 0 < K) -> K e. NN)
15	14	ancoms 484	. . . . . . 7 |- ((0 < K /\ K e. NN0) -> K e. NN)
16	10, 15	sylan 497	. . . . . 6 |- (((0 < K /\ K e. NN0) /\ (M e. NN0 /\ N e. NN0)) -> ((K x. M) gcd (K x. N)) = (K x. (M gcd N)))
17	16	anasss 488	. . . . 5 |- ((0 < K /\ (K e. NN0 /\ (M e. NN0 /\ N e. NN0))) -> ((K x. M) gcd (K x. N)) = (K x. (M gcd N)))
18	17	expcom 403	. . . 4 |- ((K e. NN0 /\ (M e. NN0 /\ N e. NN0)) -> (0 < K -> ((K x. M) gcd (K x. N)) = (K x. (M gcd N))))
19		nn0cn 7318	. . . . . . . . . . . 12 |- (M e. NN0 -> M e. CC)
20		mul02 6607	. . . . . . . . . . . 12 |- (M e. CC -> (0 x. M) = 0)
21	19, 20	syl 12	. . . . . . . . . . 11 |- (M e. NN0 -> (0 x. M) = 0)
22		opreq1 4889	. . . . . . . . . . 11 |- (0 = K -> (0 x. M) = (K x. M))
23	21, 22	sylan9req 1950	. . . . . . . . . 10 |- ((M e. NN0 /\ 0 = K) -> 0 = (K x. M))
24		nn0cn 7318	. . . . . . . . . . . 12 |- (N e. NN0 -> N e. CC)
25		mul02 6607	. . . . . . . . . . . 12 |- (N e. CC -> (0 x. N) = 0)
26	24, 25	syl 12	. . . . . . . . . . 11 |- (N e. NN0 -> (0 x. N) = 0)
27		opreq1 4889	. . . . . . . . . . 11 |- (0 = K -> (0 x. N) = (K x. N))
28	26, 27	sylan9req 1950	. . . . . . . . . 10 |- ((N e. NN0 /\ 0 = K) -> 0 = (K x. N))
29	23, 28	opreqan12d 4902	. . . . . . . . 9 |- (((M e. NN0 /\ 0 = K) /\ (N e. NN0 /\ 0 = K)) -> (0 gcd 0) = ((K x. M) gcd (K x. N)))
30	29	anandirs 571	. . . . . . . 8 |- (((M e. NN0 /\ N e. NN0) /\ 0 = K) -> (0 gcd 0) = ((K x. M) gcd (K x. N)))
31		gcd0val 13716	. . . . . . . 8 |- (0 gcd 0) = 0
32	30, 31	syl5reqr 1943	. . . . . . 7 |- (((M e. NN0 /\ N e. NN0) /\ 0 = K) -> ((K x. M) gcd (K x. N)) = 0)
33		gcdcl 13724	. . . . . . . . . 10 |- ((M e. ZZ /\ N e. ZZ) -> (M gcd N) e. NN0)
34		nn0cn 7318	. . . . . . . . . 10 |- ((M gcd N) e. NN0 -> (M gcd N) e. CC)
35		mul02 6607	. . . . . . . . . 10 |- ((M gcd N) e. CC -> (0 x. (M gcd N)) = 0)
36	33, 34, 35	3syl 24	. . . . . . . . 9 |- ((M e. ZZ /\ N e. ZZ) -> (0 x. (M gcd N)) = 0)
37		nn0z 7363	. . . . . . . . 9 |- (M e. NN0 -> M e. ZZ)
38		nn0z 7363	. . . . . . . . 9 |- (N e. NN0 -> N e. ZZ)
39	36, 37, 38	syl2an 503	. . . . . . . 8 |- ((M e. NN0 /\ N e. NN0) -> (0 x. (M gcd N)) = 0)
40		opreq1 4889	. . . . . . . 8 |- (0 = K -> (0 x. (M gcd N)) = (K x. (M gcd N)))
41	39, 40	sylan9req 1950	. . . . . . 7 |- (((M e. NN0 /\ N e. NN0) /\ 0 = K) -> 0 = (K x. (M gcd N)))
42	32, 41	eqtrd 1925	. . . . . 6 |- (((M e. NN0 /\ N e. NN0) /\ 0 = K) -> ((K x. M) gcd (K x. N)) = (K x. (M gcd N)))
43	42	ex 402	. . . . 5 |- ((M e. NN0 /\ N e. NN0) -> (0 = K -> ((K x. M) gcd (K x. N)) = (K x. (M gcd N))))
44	43	adantl 424	. . . 4 |- ((K e. NN0 /\ (M e. NN0 /\ N e. NN0)) -> (0 = K -> ((K x. M) gcd (K x. N)) = (K x. (M gcd N))))
45	18, 44	jaod 469	. . 3 |- ((K e. NN0 /\ (M e. NN0 /\ N e. NN0)) -> ((0 < K \/ 0 = K) -> ((K x. M) gcd (K x. N)) = (K x. (M gcd N))))
46	8, 45	mpd 29	. 2 |- ((K e. NN0 /\ (M e. NN0 /\ N e. NN0)) -> ((K x. M) gcd (K x. N)) = (K x. (M gcd N)))
47	46	3impb 1063	1 |- ((K e. NN0 /\ M e. NN0 /\ N e. NN0) -> ((K x. M) gcd (K x. N)) = (K x. (M gcd N)))

Syntax hints:   -> wi 3   <-> wb 163   \/ 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   x. cmul 6391   <_ cle 6448  NNcn 6449  NN0cn0 6450  ZZcz 6451   < clt 6653   gcd cgcd 13713

This theorem is referenced by:  mulgcd 13763

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-rp 7232  df-n0 7309  df-z 7345  df-fl 7463  df-mod 7500  df-uz 7587  df-seq1 7721  df-shft 7754  df-seqz 7776  df-seq0 7777  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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