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Theorem algcvgblem 15290
Description: Lemma for algcvgb 15291. (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
StepHypRef Expression
1 imor 428 . . . . 5 |- ( ( N =/= 0 -> N < M ) <-> ( -. N =/= 0 \/ N < M ) )
2 0re 10040 . . . . . . . . . . . . 13 |- 0 e. RR
3 nn0re 11301 . . . . . . . . . . . . . 14 |- ( M e. NN0 -> M e. RR )
43adantr 481 . . . . . . . . . . . . 13 |- ( ( M e. NN0 /\ N e. NN0 ) -> M e. RR )
5 ltnle 10117 . . . . . . . . . . . . 13 |- ( ( 0 e. RR /\ M e. RR ) -> ( 0 < M <-> -. M <_ 0 ) )
62, 4, 5sylancr 695 . . . . . . . . . . . 12 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 0 < M <-> -. M <_ 0 ) )
7 nn0le0eq0 11321 . . . . . . . . . . . . . 14 |- ( M e. NN0 -> ( M <_ 0 <-> M = 0 ) )
87notbid 308 . . . . . . . . . . . . 13 |- ( M e. NN0 -> ( -. M <_ 0 <-> -. M = 0 ) )
98adantr 481 . . . . . . . . . . . 12 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( -. M <_ 0 <-> -. M = 0 ) )
106, 9bitrd 268 . . . . . . . . . . 11 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 0 < M <-> -. M = 0 ) )
11 df-ne 2795 . . . . . . . . . . 11 |- ( M =/= 0 <-> -. M = 0 )
1210, 11syl6bbr 278 . . . . . . . . . 10 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 0 < M <-> M =/= 0 ) )
1312anbi2d 740 . . . . . . . . 9 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( -. N =/= 0 /\ 0 < M ) <-> ( -. N =/= 0 /\ M =/= 0 ) ) )
14 nne 2798 . . . . . . . . . . 11 |- ( -. N =/= 0 <-> N = 0 )
15 breq1 4656 . . . . . . . . . . 11 |- ( N = 0 -> ( N < M <-> 0 < M ) )
1614, 15sylbi 207 . . . . . . . . . 10 |- ( -. N =/= 0 -> ( N < M <-> 0 < M ) )
1716biimpar 502 . . . . . . . . 9 |- ( ( -. N =/= 0 /\ 0 < M ) -> N < M )
1813, 17syl6bir 244 . . . . . . . 8 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( -. N =/= 0 /\ M =/= 0 ) -> N < M ) )
1918expd 452 . . . . . . 7 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( -. N =/= 0 -> ( M =/= 0 -> N < M ) ) )
20 ax-1 6 . . . . . . 7 |- ( N < M -> ( M =/= 0 -> N < M ) )
2119, 20jctir 561 . . . . . 6 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( -. N =/= 0 -> ( M =/= 0 -> N < M ) ) /\ ( N < M -> ( M =/= 0 -> N < M ) ) ) )
22 jaob 822 . . . . . 6 |- ( ( ( -. N =/= 0 \/ N < M ) -> ( M =/= 0 -> N < M ) ) <-> ( ( -. N =/= 0 -> ( M =/= 0 -> N < M ) ) /\ ( N < M -> ( M =/= 0 -> N < M ) ) ) )
2321, 22sylibr 224 . . . . 5 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( -. N =/= 0 \/ N < M ) -> ( M =/= 0 -> N < M ) ) )
241, 23syl5bi 232 . . . 4 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( N =/= 0 -> N < M ) -> ( M =/= 0 -> N < M ) ) )
25 nn0ge0 11318 . . . . . . . 8 |- ( N e. NN0 -> 0 <_ N )
2625adantl 482 . . . . . . 7 |- ( ( M e. NN0 /\ N e. NN0 ) -> 0 <_ N )
27 nn0re 11301 . . . . . . . 8 |- ( N e. NN0 -> N e. RR )
28 lelttr 10128 . . . . . . . . 9 |- ( ( 0 e. RR /\ N e. RR /\ M e. RR ) -> ( ( 0 <_ N /\ N < M ) -> 0 < M ) )
292, 28mp3an1 1411 . . . . . . . 8 |- ( ( N e. RR /\ M e. RR ) -> ( ( 0 <_ N /\ N < M ) -> 0 < M ) )
3027, 3, 29syl2anr 495 . . . . . . 7 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( 0 <_ N /\ N < M ) -> 0 < M ) )
3126, 30mpand 711 . . . . . 6 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( N < M -> 0 < M ) )
3231, 12sylibd 229 . . . . 5 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( N < M -> M =/= 0 ) )
3332imim2d 57 . . . 4 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( N =/= 0 -> N < M ) -> ( N =/= 0 -> M =/= 0 ) ) )
3424, 33jcad 555 . . 3 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( N =/= 0 -> N < M ) -> ( ( M =/= 0 -> N < M ) /\ ( N =/= 0 -> M =/= 0 ) ) ) )
35 pm3.34 610 . . 3 |- ( ( ( M =/= 0 -> N < M ) /\ ( N =/= 0 -> M =/= 0 ) ) -> ( N =/= 0 -> N < M ) )
3634, 35impbid1 215 . 2 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( N =/= 0 -> N < M ) <-> ( ( M =/= 0 -> N < M ) /\ ( N =/= 0 -> M =/= 0 ) ) ) )
37 con34b 306 . . . 4 |- ( ( M = 0 -> N = 0 ) <-> ( -. N = 0 -> -. M = 0 ) )
38 df-ne 2795 . . . . 5 |- ( N =/= 0 <-> -. N = 0 )
3938, 11imbi12i 340 . . . 4 |- ( ( N =/= 0 -> M =/= 0 ) <-> ( -. N = 0 -> -. M = 0 ) )
4037, 39bitr4i 267 . . 3 |- ( ( M = 0 -> N = 0 ) <-> ( N =/= 0 -> M =/= 0 ) )
4140anbi2i 730 . 2 |- ( ( ( M =/= 0 -> N < M ) /\ ( M = 0 -> N = 0 ) ) <-> ( ( M =/= 0 -> N < M ) /\ ( N =/= 0 -> M =/= 0 ) ) )
4236, 41syl6bbr 278 1 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( N =/= 0 -> N < M ) <-> ( ( M =/= 0 -> N < M ) /\ ( M = 0 -> N = 0 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   RRcr 9935   0cc0 9936   < clt 10074   <_ cle 10075   NN0cn0 11292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293
This theorem is referenced by:  algcvgb  15291
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