Theorem algcvgblem 11576

Description: Lemma for algcvgb 11577. (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 8978	. . . . . . . . 9 |- ( N e. NN0 -> N e. ZZ )
2		0z 8969	. . . . . . . . 9 |- 0 e. ZZ
3		zdceq 9030	. . . . . . . . 9 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
4	1, 2, 3	sylancl 407	. . . . . . . 8 |- ( N e. NN0 -> DECID N = 0 )
5	4	dcned 2288	. . . . . . 7 |- ( N e. NN0 -> DECID N =/= 0 )
6		imordc 865	. . . . . . 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 273	. . . . 5 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( N =/= 0 -> N < M ) <-> ( -. N =/= 0 \/ N < M ) ) )
9		nn0z 8978	. . . . . . . . . . . . . 14 |- ( M e. NN0 -> M e. ZZ )
10		zltnle 9004	. . . . . . . . . . . . . 14 |- ( ( 0 e. ZZ /\ M e. ZZ ) -> ( 0 < M <-> -. M <_ 0 ) )
11	2, 9, 10	sylancr 408	. . . . . . . . . . . . 13 |- ( M e. NN0 -> ( 0 < M <-> -. M <_ 0 ) )
12	11	adantr 272	. . . . . . . . . . . 12 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 0 < M <-> -. M <_ 0 ) )
13		nn0le0eq0 8909	. . . . . . . . . . . . . 14 |- ( M e. NN0 -> ( M <_ 0 <-> M = 0 ) )
14	13	notbid 639	. . . . . . . . . . . . 13 |- ( M e. NN0 -> ( -. M <_ 0 <-> -. M = 0 ) )
15	14	adantr 272	. . . . . . . . . . . 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 2283	. . . . . . . . . . 11 |- ( M =/= 0 <-> -. M = 0 )
18	16, 17	syl6bbr 197	. . . . . . . . . 10 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 0 < M <-> M =/= 0 ) )
19	18	anbi2d 457	. . . . . . . . 9 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( -. N =/= 0 /\ 0 < M ) <-> ( -. N =/= 0 /\ M =/= 0 ) ) )
20	1	adantl 273	. . . . . . . . . . . . . 14 |- ( ( M e. NN0 /\ N e. NN0 ) -> N e. ZZ )
21	20, 2, 3	sylancl 407	. . . . . . . . . . . . 13 |- ( ( M e. NN0 /\ N e. NN0 ) -> DECID N = 0 )
22		nnedc 2287	. . . . . . . . . . . . 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 3898	. . . . . . . . . . . 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		bi2 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 5	. . . . . . 7 |- ( N < M -> ( M =/= 0 -> N < M ) )
32	30, 31	jctir 309	. . . . . 6 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( -. N =/= 0 -> ( M =/= 0 -> N < M ) ) /\ ( N < M -> ( M =/= 0 -> N < M ) ) ) )
33		jaob 682	. . . . . 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 8906	. . . . . . . 8 |- ( N e. NN0 -> 0 <_ N )
37	36	adantl 273	. . . . . . 7 |- ( ( M e. NN0 /\ N e. NN0 ) -> 0 <_ N )
38		nn0re 8890	. . . . . . . 8 |- ( N e. NN0 -> N e. RR )
39		nn0re 8890	. . . . . . . 8 |- ( M e. NN0 -> M e. RR )
40		0re 7690	. . . . . . . . 9 |- 0 e. RR
41		lelttr 7775	. . . . . . . . 9 |- ( ( 0 e. RR /\ N e. RR /\ M e. RR ) -> ( ( 0 <_ N /\ N < M ) -> 0 < M ) )
42	40, 41	mp3an1 1285	. . . . . . . 8 |- ( ( N e. RR /\ M e. RR ) -> ( ( 0 <_ N /\ N < M ) -> 0 < M ) )
43	38, 39, 42	syl2anr 286	. . . . . . 7 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( 0 <_ N /\ N < M ) -> 0 < M ) )
44	37, 43	mpand 423	. . . . . 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 303	. . 3 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( N =/= 0 -> N < M ) -> ( ( M =/= 0 -> N < M ) /\ ( N =/= 0 -> M =/= 0 ) ) ) )
48		pm3.34 341	. . 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 839	. . . . 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 2283	. . . . 5 |- ( N =/= 0 <-> -. N = 0 )
53	52, 17	imbi12i 238	. . . 4 |- ( ( N =/= 0 -> M =/= 0 ) <-> ( -. N = 0 -> -. M = 0 ) )
54	51, 53	syl6bbr 197	. . 3 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( M = 0 -> N = 0 ) <-> ( N =/= 0 -> M =/= 0 ) ) )
55	54	anbi2d 457	. 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 680  DECID wdc 802   = wceq 1314   e. wcel 1463   =/= wne 2282   class class class wbr 3895   RRcr 7546   0cc0 7547   < clt 7724   <_ cle 7725   NN0cn0 8881   ZZcz 8958

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-addcom 7645  ax-addass 7647  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-0id 7653  ax-rnegex 7654  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-apti 7660  ax-pre-ltadd 7661

This theorem depends on definitions:  df-bi 116  df-stab 799  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-br 3896  df-opab 3950  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-iota 5046  df-fun 5083  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-inn 8631  df-n0 8882  df-z 8959

This theorem is referenced by:  algcvgb  11577