Theorem algcvgblem 12614

Description: Lemma for algcvgb 12615. (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 9492	. . . . . . . . 9 |- ( N e. NN0 -> N e. ZZ )
2		0z 9483	. . . . . . . . 9 |- 0 e. ZZ
3		zdceq 9548	. . . . . . . . 9 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
4	1, 2, 3	sylancl 413	. . . . . . . 8 |- ( N e. NN0 -> DECID N = 0 )
5	4	dcned 2406	. . . . . . 7 |- ( N e. NN0 -> DECID N =/= 0 )
6		imordc 902	. . . . . . 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 277	. . . . 5 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( N =/= 0 -> N < M ) <-> ( -. N =/= 0 \/ N < M ) ) )
9		nn0z 9492	. . . . . . . . . . . . . 14 |- ( M e. NN0 -> M e. ZZ )
10		zltnle 9518	. . . . . . . . . . . . . 14 |- ( ( 0 e. ZZ /\ M e. ZZ ) -> ( 0 < M <-> -. M <_ 0 ) )
11	2, 9, 10	sylancr 414	. . . . . . . . . . . . 13 |- ( M e. NN0 -> ( 0 < M <-> -. M <_ 0 ) )
12	11	adantr 276	. . . . . . . . . . . 12 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 0 < M <-> -. M <_ 0 ) )
13		nn0le0eq0 9423	. . . . . . . . . . . . . 14 |- ( M e. NN0 -> ( M <_ 0 <-> M = 0 ) )
14	13	notbid 671	. . . . . . . . . . . . 13 |- ( M e. NN0 -> ( -. M <_ 0 <-> -. M = 0 ) )
15	14	adantr 276	. . . . . . . . . . . 12 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( -. M <_ 0 <-> -. M = 0 ) )
16	12, 15	bitrd 188	. . . . . . . . . . 11 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 0 < M <-> -. M = 0 ) )
17		df-ne 2401	. . . . . . . . . . 11 |- ( M =/= 0 <-> -. M = 0 )
18	16, 17	bitr4di 198	. . . . . . . . . 10 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 0 < M <-> M =/= 0 ) )
19	18	anbi2d 464	. . . . . . . . 9 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( -. N =/= 0 /\ 0 < M ) <-> ( -. N =/= 0 /\ M =/= 0 ) ) )
20	1	adantl 277	. . . . . . . . . . . . . 14 |- ( ( M e. NN0 /\ N e. NN0 ) -> N e. ZZ )
21	20, 2, 3	sylancl 413	. . . . . . . . . . . . 13 |- ( ( M e. NN0 /\ N e. NN0 ) -> DECID N = 0 )
22		nnedc 2405	. . . . . . . . . . . . 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 4089	. . . . . . . . . . . 12 |- ( N = 0 -> ( N < M <-> 0 < M ) )
25	23, 24	biimtrdi 163	. . . . . . . . . . 11 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( -. N =/= 0 -> ( N < M <-> 0 < M ) ) )
26		biimpr 130	. . . . . . . . . . 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 254	. . . . . . . . 9 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( -. N =/= 0 /\ 0 < M ) -> N < M ) )
29	19, 28	sylbird 170	. . . . . . . 8 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( -. N =/= 0 /\ M =/= 0 ) -> N < M ) )
30	29	expd 258	. . . . . . 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 313	. . . . . 6 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( -. N =/= 0 -> ( M =/= 0 -> N < M ) ) /\ ( N < M -> ( M =/= 0 -> N < M ) ) ) )
33		jaob 715	. . . . . 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 134	. . . . 5 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( -. N =/= 0 \/ N < M ) -> ( M =/= 0 -> N < M ) ) )
35	8, 34	sylbid 150	. . . 4 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( N =/= 0 -> N < M ) -> ( M =/= 0 -> N < M ) ) )
36		nn0ge0 9420	. . . . . . . 8 |- ( N e. NN0 -> 0 <_ N )
37	36	adantl 277	. . . . . . 7 |- ( ( M e. NN0 /\ N e. NN0 ) -> 0 <_ N )
38		nn0re 9404	. . . . . . . 8 |- ( N e. NN0 -> N e. RR )
39		nn0re 9404	. . . . . . . 8 |- ( M e. NN0 -> M e. RR )
40		0re 8172	. . . . . . . . 9 |- 0 e. RR
41		lelttr 8261	. . . . . . . . 9 |- ( ( 0 e. RR /\ N e. RR /\ M e. RR ) -> ( ( 0 <_ N /\ N < M ) -> 0 < M ) )
42	40, 41	mp3an1 1358	. . . . . . . 8 |- ( ( N e. RR /\ M e. RR ) -> ( ( 0 <_ N /\ N < M ) -> 0 < M ) )
43	38, 39, 42	syl2anr 290	. . . . . . 7 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( 0 <_ N /\ N < M ) -> 0 < M ) )
44	37, 43	mpand 429	. . . . . 6 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( N < M -> 0 < M ) )
45	44, 18	sylibd 149	. . . . 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 307	. . 3 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( N =/= 0 -> N < M ) -> ( ( M =/= 0 -> N < M ) /\ ( N =/= 0 -> M =/= 0 ) ) ) )
48		pm3.34 346	. . 3 |- ( ( ( M =/= 0 -> N < M ) /\ ( N =/= 0 -> M =/= 0 ) ) -> ( N =/= 0 -> N < M ) )
49	47, 48	impbid1 142	. 2 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( N =/= 0 -> N < M ) <-> ( ( M =/= 0 -> N < M ) /\ ( N =/= 0 -> M =/= 0 ) ) ) )
50		con34bdc 876	. . . . 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 2401	. . . . 5 |- ( N =/= 0 <-> -. N = 0 )
53	52, 17	imbi12i 239	. . . 4 |- ( ( N =/= 0 -> M =/= 0 ) <-> ( -. N = 0 -> -. M = 0 ) )
54	51, 53	bitr4di 198	. . 3 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( M = 0 -> N = 0 ) <-> ( N =/= 0 -> M =/= 0 ) ) )
55	54	anbi2d 464	. 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 191	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 104   <-> wb 105   \/ wo 713  DECID wdc 839   = wceq 1395   e. wcel 2200   =/= wne 2400   class class class wbr 4086   RRcr 8024   0cc0 8025   < clt 8207   <_ cle 8208   NN0cn0 9395   ZZcz 9472

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-addcom 8125  ax-addass 8127  ax-distr 8129  ax-i2m1 8130  ax-0lt1 8131  ax-0id 8133  ax-rnegex 8134  ax-cnre 8136  ax-pre-ltirr 8137  ax-pre-ltwlin 8138  ax-pre-lttrn 8139  ax-pre-apti 8140  ax-pre-ltadd 8141

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-iota 5284  df-fun 5326  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8209  df-mnf 8210  df-xr 8211  df-ltxr 8212  df-le 8213  df-sub 8345  df-neg 8346  df-inn 9137  df-n0 9396  df-z 9473

This theorem is referenced by:  algcvgb  12615