Theorem algcvgblem 11981

Description: Lemma for algcvgb 11982. (Contributed by Paul Chapman, 31-Mar-2011.)

Assertion

Ref	Expression
algcvgblem	|- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( N =/= 0 -> N < M ) <-> ( ( M =/= 0 -> N < M ) /\ ( M = 0 -> N = 0 ) ) ) )

Proof of Theorem algcvgblem

Step	Hyp	Ref	Expression
1		nn0z 9211	. . . . . . . . 9 |- ( N e. NN0 -> N e. ZZ )
2		0z 9202	. . . . . . . . 9 |- 0 e. ZZ
3		zdceq 9266	. . . . . . . . 9 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
4	1, 2, 3	sylancl 410	. . . . . . . 8 |- ( N e. NN0 -> DECID N = 0 )
5	4	dcned 2342	. . . . . . 7 |- ( N e. NN0 -> DECID N =/= 0 )
6		imordc 887	. . . . . . 7 |- (DECID N =/= 0 -> ( ( N =/= 0 -> N < M ) <-> ( -. N =/= 0 \/ N < M ) ) )
7	5, 6	syl 14	. . . . . 6 |- ( N e. NN0 -> ( ( N =/= 0 -> N < M ) <-> ( -. N =/= 0 \/ N < M ) ) )
8	7	adantl 275	. . . . 5 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( N =/= 0 -> N < M ) <-> ( -. N =/= 0 \/ N < M ) ) )
9		nn0z 9211	. . . . . . . . . . . . . 14 |- ( M e. NN0 -> M e. ZZ )
10		zltnle 9237	. . . . . . . . . . . . . 14 |- ( ( 0 e. ZZ /\ M e. ZZ ) -> ( 0 < M <-> -. M <_ 0 ) )
11	2, 9, 10	sylancr 411	. . . . . . . . . . . . 13 |- ( M e. NN0 -> ( 0 < M <-> -. M <_ 0 ) )
12	11	adantr 274	. . . . . . . . . . . 12 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 0 < M <-> -. M <_ 0 ) )
13		nn0le0eq0 9142	. . . . . . . . . . . . . 14 |- ( M e. NN0 -> ( M <_ 0 <-> M = 0 ) )
14	13	notbid 657	. . . . . . . . . . . . 13 |- ( M e. NN0 -> ( -. M <_ 0 <-> -. M = 0 ) )
15	14	adantr 274	. . . . . . . . . . . 12 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( -. M <_ 0 <-> -. M = 0 ) )
16	12, 15	bitrd 187	. . . . . . . . . . 11 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 0 < M <-> -. M = 0 ) )
17		df-ne 2337	. . . . . . . . . . 11 |- ( M =/= 0 <-> -. M = 0 )
18	16, 17	bitr4di 197	. . . . . . . . . 10 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 0 < M <-> M =/= 0 ) )
19	18	anbi2d 460	. . . . . . . . 9 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( -. N =/= 0 /\ 0 < M ) <-> ( -. N =/= 0 /\ M =/= 0 ) ) )
20	1	adantl 275	. . . . . . . . . . . . . 14 |- ( ( M e. NN0 /\ N e. NN0 ) -> N e. ZZ )
21	20, 2, 3	sylancl 410	. . . . . . . . . . . . 13 |- ( ( M e. NN0 /\ N e. NN0 ) -> DECID N = 0 )
22		nnedc 2341	. . . . . . . . . . . . 13 |- (DECID N = 0 -> ( -. N =/= 0 <-> N = 0 ) )
23	21, 22	syl 14	. . . . . . . . . . . 12 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( -. N =/= 0 <-> N = 0 ) )
24		breq1 3985	. . . . . . . . . . . 12 |- ( N = 0 -> ( N < M <-> 0 < M ) )
25	23, 24	syl6bi 162	. . . . . . . . . . 11 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( -. N =/= 0 -> ( N < M <-> 0 < M ) ) )
26		biimpr 129	. . . . . . . . . . 11 |- ( ( N < M <-> 0 < M ) -> ( 0 < M -> N < M ) )
27	25, 26	syl6 33	. . . . . . . . . 10 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( -. N =/= 0 -> ( 0 < M -> N < M ) ) )
28	27	impd 252	. . . . . . . . 9 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( -. N =/= 0 /\ 0 < M ) -> N < M ) )
29	19, 28	sylbird 169	. . . . . . . 8 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( -. N =/= 0 /\ M =/= 0 ) -> N < M ) )
30	29	expd 256	. . . . . . 7 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( -. N =/= 0 -> ( M =/= 0 -> N < M ) ) )
31		ax-1 6	. . . . . . 7 |- ( N < M -> ( M =/= 0 -> N < M ) )
32	30, 31	jctir 311	. . . . . 6 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( -. N =/= 0 -> ( M =/= 0 -> N < M ) ) /\ ( N < M -> ( M =/= 0 -> N < M ) ) ) )
33		jaob 700	. . . . . 6 |- ( ( ( -. N =/= 0 \/ N < M ) -> ( M =/= 0 -> N < M ) ) <-> ( ( -. N =/= 0 -> ( M =/= 0 -> N < M ) ) /\ ( N < M -> ( M =/= 0 -> N < M ) ) ) )
34	32, 33	sylibr 133	. . . . 5 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( -. N =/= 0 \/ N < M ) -> ( M =/= 0 -> N < M ) ) )
35	8, 34	sylbid 149	. . . 4 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( N =/= 0 -> N < M ) -> ( M =/= 0 -> N < M ) ) )
36		nn0ge0 9139	. . . . . . . 8 |- ( N e. NN0 -> 0 <_ N )
37	36	adantl 275	. . . . . . 7 |- ( ( M e. NN0 /\ N e. NN0 ) -> 0 <_ N )
38		nn0re 9123	. . . . . . . 8 |- ( N e. NN0 -> N e. RR )
39		nn0re 9123	. . . . . . . 8 |- ( M e. NN0 -> M e. RR )
40		0re 7899	. . . . . . . . 9 |- 0 e. RR
41		lelttr 7987	. . . . . . . . 9 |- ( ( 0 e. RR /\ N e. RR /\ M e. RR ) -> ( ( 0 <_ N /\ N < M ) -> 0 < M ) )
42	40, 41	mp3an1 1314	. . . . . . . 8 |- ( ( N e. RR /\ M e. RR ) -> ( ( 0 <_ N /\ N < M ) -> 0 < M ) )
43	38, 39, 42	syl2anr 288	. . . . . . 7 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( 0 <_ N /\ N < M ) -> 0 < M ) )
44	37, 43	mpand 426	. . . . . 6 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( N < M -> 0 < M ) )
45	44, 18	sylibd 148	. . . . 5 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( N < M -> M =/= 0 ) )
46	45	imim2d 54	. . . 4 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( N =/= 0 -> N < M ) -> ( N =/= 0 -> M =/= 0 ) ) )
47	35, 46	jcad 305	. . 3 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( N =/= 0 -> N < M ) -> ( ( M =/= 0 -> N < M ) /\ ( N =/= 0 -> M =/= 0 ) ) ) )
48		pm3.34 344	. . 3 |- ( ( ( M =/= 0 -> N < M ) /\ ( N =/= 0 -> M =/= 0 ) ) -> ( N =/= 0 -> N < M ) )
49	47, 48	impbid1 141	. 2 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( N =/= 0 -> N < M ) <-> ( ( M =/= 0 -> N < M ) /\ ( N =/= 0 -> M =/= 0 ) ) ) )
50		con34bdc 861	. . . . 5 |- (DECID N = 0 -> ( ( M = 0 -> N = 0 ) <-> ( -. N = 0 -> -. M = 0 ) ) )
51	21, 50	syl 14	. . . 4 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( M = 0 -> N = 0 ) <-> ( -. N = 0 -> -. M = 0 ) ) )
52		df-ne 2337	. . . . 5 |- ( N =/= 0 <-> -. N = 0 )
53	52, 17	imbi12i 238	. . . 4 |- ( ( N =/= 0 -> M =/= 0 ) <-> ( -. N = 0 -> -. M = 0 ) )
54	51, 53	bitr4di 197	. . 3 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( M = 0 -> N = 0 ) <-> ( N =/= 0 -> M =/= 0 ) ) )
55	54	anbi2d 460	. 2 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( ( M =/= 0 -> N < M ) /\ ( M = 0 -> N = 0 ) ) <-> ( ( M =/= 0 -> N < M ) /\ ( N =/= 0 -> M =/= 0 ) ) ) )
56	49, 55	bitr4d 190	1 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( N =/= 0 -> N < M ) <-> ( ( M =/= 0 -> N < M ) /\ ( M = 0 -> N = 0 ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698  DECID wdc 824   = wceq 1343   e. wcel 2136   =/= wne 2336   class class class wbr 3982   RRcr 7752   0cc0 7753   < clt 7933   <_ cle 7934   NN0cn0 9114   ZZcz 9191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-n0 9115  df-z 9192

This theorem is referenced by:  algcvgb  11982