Theorem eucalglt 13753

Description: The second member of the state decreases with each iteration of the step function E for Euclid's Algorithm. (Contributed by Paul Chapman, 31-Mar-2011.)

Hypothesis

Ref	Expression
eucalgval.1	|- E = {<.x, y>. | (x e. (NN0 X. NN0) /\ y = if((2nd` x) = 0, x, <.(2nd` x), ( mod ` x)>.))}

Assertion

Ref	Expression
eucalglt	|- (X e. (NN0 X. NN0) -> ((2nd` (E` X)) =/= 0 -> (2nd` (E` X)) < (2nd` X)))

Distinct variable group:   x,y

Proof of Theorem eucalglt

Step	Hyp	Ref	Expression
1		elxp2 4019	. 2 |- (X e. (NN0 X. NN0) <-> E.m e. NN0 E.n e. NN0 X = <.m, n>.)
2		fveq2 4681	. . . . . . 7 |- (X = <.m, n>. -> (E` X) = (E` <.m, n>.))
3	2	fveq2d 4685	. . . . . 6 |- (X = <.m, n>. -> (2nd` (E` X)) = (2nd` (E` <.m, n>.)))
4	3	neeq1d 2028	. . . . 5 |- (X = <.m, n>. -> ((2nd` (E` X)) =/= 0 <-> (2nd` (E` <.m, n>.)) =/= 0))
5		fveq2 4681	. . . . . 6 |- (X = <.m, n>. -> (2nd` X) = (2nd` <.m, n>.))
6	3, 5	breq12d 3351	. . . . 5 |- (X = <.m, n>. -> ((2nd` (E` X)) < (2nd` X) <-> (2nd` (E` <.m, n>.)) < (2nd` <.m, n>.)))
7	4, 6	imbi12d 688	. . . 4 |- (X = <.m, n>. -> (((2nd` (E` X)) =/= 0 -> (2nd` (E` X)) < (2nd` X)) <-> ((2nd` (E` <.m, n>.)) =/= 0 -> (2nd` (E` <.m, n>.)) < (2nd` <.m, n>.))))
8		pm2.24 95	. . . . . . . 8 |- (n = 0 -> (-. n = 0 -> n < n))
9		df-ne 2019	. . . . . . . 8 |- (n =/= 0 <-> -. n = 0)
10	8, 9	syl5ib 223	. . . . . . 7 |- (n = 0 -> (n =/= 0 -> n < n))
11	10	adantl 424	. . . . . 6 |- (((m e. NN0 /\ n e. NN0) /\ n = 0) -> (n =/= 0 -> n < n))
12		eucalgval.1	. . . . . . . . . . . 12 |- E = {<.x, y>. | (x e. (NN0 X. NN0) /\ y = if((2nd` x) = 0, x, <.(2nd` x), ( mod ` x)>.))}
13	12	eucalgval2 13750	. . . . . . . . . . 11 |- ((m e. NN0 /\ n e. NN0) -> (E` <.m, n>.) = if(n = 0, <.m, n>., <.n, (m mod n)>.))
14		iftrue 2989	. . . . . . . . . . 11 |- (n = 0 -> if(n = 0, <.m, n>., <.n, (m mod n)>.) = <.m, n>.)
15	13, 14	sylan9eq 1948	. . . . . . . . . 10 |- (((m e. NN0 /\ n e. NN0) /\ n = 0) -> (E` <.m, n>.) = <.m, n>.)
16	15	fveq2d 4685	. . . . . . . . 9 |- (((m e. NN0 /\ n e. NN0) /\ n = 0) -> (2nd` (E` <.m, n>.)) = (2nd` <.m, n>.))
17		visset 2295	. . . . . . . . . 10 |- m e. _V
18		visset 2295	. . . . . . . . . 10 |- n e. _V
19	17, 18	op2nd 5027	. . . . . . . . 9 |- (2nd` <.m, n>.) = n
20	16, 19	syl6eq 1944	. . . . . . . 8 |- (((m e. NN0 /\ n e. NN0) /\ n = 0) -> (2nd` (E` <.m, n>.)) = n)
21	20	neeq1d 2028	. . . . . . 7 |- (((m e. NN0 /\ n e. NN0) /\ n = 0) -> ((2nd` (E` <.m, n>.)) =/= 0 <-> n =/= 0))
22	19	a1i 8	. . . . . . . 8 |- (((m e. NN0 /\ n e. NN0) /\ n = 0) -> (2nd` <.m, n>.) = n)
23	20, 22	breq12d 3351	. . . . . . 7 |- (((m e. NN0 /\ n e. NN0) /\ n = 0) -> ((2nd` (E` <.m, n>.)) < (2nd` <.m, n>.) <-> n < n))
24	21, 23	imbi12d 688	. . . . . 6 |- (((m e. NN0 /\ n e. NN0) /\ n = 0) -> (((2nd` (E` <.m, n>.)) =/= 0 -> (2nd` (E` <.m, n>.)) < (2nd` <.m, n>.)) <-> (n =/= 0 -> n < n)))
25	11, 24	mpbird 213	. . . . 5 |- (((m e. NN0 /\ n e. NN0) /\ n = 0) -> ((2nd` (E` <.m, n>.)) =/= 0 -> (2nd` (E` <.m, n>.)) < (2nd` <.m, n>.)))
26		modlt 7504	. . . . . . . . . . 11 |- ((m e. RR /\ n e. RR+) -> (m mod n) < n)
27		nn0re 7317	. . . . . . . . . . 11 |- (m e. NN0 -> m e. RR)
28		nnrp 7238	. . . . . . . . . . 11 |- (n e. NN -> n e. RR+)
29	26, 27, 28	syl2an 503	. . . . . . . . . 10 |- ((m e. NN0 /\ n e. NN) -> (m mod n) < n)
30	29	adantrr 431	. . . . . . . . 9 |- ((m e. NN0 /\ (n e. NN /\ -. n = 0)) -> (m mod n) < n)
31		elnn0 7310	. . . . . . . . . . 11 |- (n e. NN0 <-> (n e. NN \/ n = 0))
32	31	anbi1i 539	. . . . . . . . . 10 |- ((n e. NN0 /\ -. n = 0) <-> ((n e. NN \/ n = 0) /\ -. n = 0))
33		pm5.61 496	. . . . . . . . . 10 |- (((n e. NN \/ n = 0) /\ -. n = 0) <-> (n e. NN /\ -. n = 0))
34	32, 33	bitri 190	. . . . . . . . 9 |- ((n e. NN0 /\ -. n = 0) <-> (n e. NN /\ -. n = 0))
35	30, 34	sylan2b 501	. . . . . . . 8 |- ((m e. NN0 /\ (n e. NN0 /\ -. n = 0)) -> (m mod n) < n)
36	35	anassrs 489	. . . . . . 7 |- (((m e. NN0 /\ n e. NN0) /\ -. n = 0) -> (m mod n) < n)
37		iffalse 2991	. . . . . . . . . . 11 |- (-. n = 0 -> if(n = 0, <.m, n>., <.n, (m mod n)>.) = <.n, (m mod n)>.)
38	13, 37	sylan9eq 1948	. . . . . . . . . 10 |- (((m e. NN0 /\ n e. NN0) /\ -. n = 0) -> (E` <.m, n>.) = <.n, (m mod n)>.)
39	38	fveq2d 4685	. . . . . . . . 9 |- (((m e. NN0 /\ n e. NN0) /\ -. n = 0) -> (2nd` (E` <.m, n>.)) = (2nd` <.n, (m mod n)>.))
40		oprex 4907	. . . . . . . . . 10 |- (m mod n) e. _V
41	18, 40	op2nd 5027	. . . . . . . . 9 |- (2nd` <.n, (m mod n)>.) = (m mod n)
42	39, 41	syl6eq 1944	. . . . . . . 8 |- (((m e. NN0 /\ n e. NN0) /\ -. n = 0) -> (2nd` (E` <.m, n>.)) = (m mod n))
43	19	a1i 8	. . . . . . . 8 |- (((m e. NN0 /\ n e. NN0) /\ -. n = 0) -> (2nd` <.m, n>.) = n)
44	42, 43	breq12d 3351	. . . . . . 7 |- (((m e. NN0 /\ n e. NN0) /\ -. n = 0) -> ((2nd` (E` <.m, n>.)) < (2nd` <.m, n>.) <-> (m mod n) < n))
45	36, 44	mpbird 213	. . . . . 6 |- (((m e. NN0 /\ n e. NN0) /\ -. n = 0) -> (2nd` (E` <.m, n>.)) < (2nd` <.m, n>.))
46	45	a1d 15	. . . . 5 |- (((m e. NN0 /\ n e. NN0) /\ -. n = 0) -> ((2nd` (E` <.m, n>.)) =/= 0 -> (2nd` (E` <.m, n>.)) < (2nd` <.m, n>.)))
47	25, 46	pm2.61dan 535	. . . 4 |- ((m e. NN0 /\ n e. NN0) -> ((2nd` (E` <.m, n>.)) =/= 0 -> (2nd` (E` <.m, n>.)) < (2nd` <.m, n>.)))
48	7, 47	syl5cbir 228	. . 3 |- ((m e. NN0 /\ n e. NN0) -> (X = <.m, n>. -> ((2nd` (E` X)) =/= 0 -> (2nd` (E` X)) < (2nd` X))))
49	48	r19.23aivv 2217	. 2 |- (E.m e. NN0 E.n e. NN0 X = <.m, n>. -> ((2nd` (E` X)) =/= 0 -> (2nd` (E` X)) < (2nd` X)))
50	1, 49	sylbi 216	1 |- (X e. (NN0 X. NN0) -> ((2nd` (E` X)) =/= 0 -> (2nd` (E` X)) < (2nd` X)))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  E.wrex 2106  ifcif 2982  <.cop 3046   class class class wbr 3338  {copab 3395   X. cxp 3984  ` cfv 3998  (class class class)co 4884  2ndc2nd 5019  RRcr 6385  0cc0 6386  NNcn 6449  NN0cn0 6450  RR+crp 6453   < clt 6653   mod cmo 7499

This theorem is referenced by:  eucalgcvga 13754  mulgcdlem5 13760

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-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-rp 7232  df-n0 7309  df-z 7345  df-fl 7463  df-mod 7500

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