Theorem algcvgblem 12682

Description: Lemma for algcvgb 12683. (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 9542	. . . . . . . . 9 |- ( N e. NN0 -> N e. ZZ )
2		0z 9533	. . . . . . . . 9 |- 0 e. ZZ
3		zdceq 9598	. . . . . . . . 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 2409	. . . . . . 7 |- ( N e. NN0 -> DECID N =/= 0 )
6		imordc 905	. . . . . . 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 9542	. . . . . . . . . . . . . 14 |- ( M e. NN0 -> M e. ZZ )
10		zltnle 9568	. . . . . . . . . . . . . 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 9473	. . . . . . . . . . . . . 14 |- ( M e. NN0 -> ( M <_ 0 <-> M = 0 ) )
14	13	notbid 673	. . . . . . . . . . . . 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 2404	. . . . . . . . . . 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 2408	. . . . . . . . . . . . 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 4096	. . . . . . . . . . . 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 718	. . . . . 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 9470	. . . . . . . 8 |- ( N e. NN0 -> 0 <_ N )
37	36	adantl 277	. . . . . . 7 |- ( ( M e. NN0 /\ N e. NN0 ) -> 0 <_ N )
38		nn0re 9454	. . . . . . . 8 |- ( N e. NN0 -> N e. RR )
39		nn0re 9454	. . . . . . . 8 |- ( M e. NN0 -> M e. RR )
40		0re 8222	. . . . . . . . 9 |- 0 e. RR
41		lelttr 8311	. . . . . . . . 9 |- ( ( 0 e. RR /\ N e. RR /\ M e. RR ) -> ( ( 0 <_ N /\ N < M ) -> 0 < M ) )
42	40, 41	mp3an1 1361	. . . . . . . 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 879	. . . . 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 2404	. . . . 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 716  DECID wdc 842   = wceq 1398   e. wcel 2202   =/= wne 2403   class class class wbr 4093   RRcr 8074   0cc0 8075   < clt 8257   <_ cle 8258   NN0cn0 9445   ZZcz 9522

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-inn 9187  df-n0 9446  df-z 9523

This theorem is referenced by:  algcvgb  12683