Theorem zaddcllemneg 9662

Description: Lemma for zaddcl 9663. Special case in which -u N is a positive integer. (Contributed by Jim Kingdon, 14-Mar-2020.)

Assertion

Ref	Expression
zaddcllemneg	|- ( ( M e. ZZ /\ N e. RR /\ -u N e. NN ) -> ( M + N ) e. ZZ )

Proof of Theorem zaddcllemneg

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1029	. . . . 5 |- ( ( M e. ZZ /\ N e. RR /\ -u N e. NN ) -> N e. RR )
2	1	recnd 8344	. . . 4 |- ( ( M e. ZZ /\ N e. RR /\ -u N e. NN ) -> N e. CC )
3	2	negnegd 8618	. . 3 |- ( ( M e. ZZ /\ N e. RR /\ -u N e. NN ) -> -u -u N = N )
4	3	oveq2d 6091	. 2 |- ( ( M e. ZZ /\ N e. RR /\ -u N e. NN ) -> ( M + -u -u N ) = ( M + N ) )
5		negeq 8509	. . . . . . . 8 |- ( x = 1 -> -u x = -u 1 )
6	5	oveq2d 6091	. . . . . . 7 |- ( x = 1 -> ( M + -u x ) = ( M + -u 1 ) )
7	6	eleq1d 2307	. . . . . 6 |- ( x = 1 -> ( ( M + -u x ) e. ZZ <-> ( M + -u 1 ) e. ZZ ) )
8	7	imbi2d 230	. . . . 5 |- ( x = 1 -> ( ( ( M e. ZZ /\ N e. RR ) -> ( M + -u x ) e. ZZ ) <-> ( ( M e. ZZ /\ N e. RR ) -> ( M + -u 1 ) e. ZZ ) ) )
9		negeq 8509	. . . . . . . 8 |- ( x = y -> -u x = -u y )
10	9	oveq2d 6091	. . . . . . 7 |- ( x = y -> ( M + -u x ) = ( M + -u y ) )
11	10	eleq1d 2307	. . . . . 6 |- ( x = y -> ( ( M + -u x ) e. ZZ <-> ( M + -u y ) e. ZZ ) )
12	11	imbi2d 230	. . . . 5 |- ( x = y -> ( ( ( M e. ZZ /\ N e. RR ) -> ( M + -u x ) e. ZZ ) <-> ( ( M e. ZZ /\ N e. RR ) -> ( M + -u y ) e. ZZ ) ) )
13		negeq 8509	. . . . . . . 8 |- ( x = ( y + 1 ) -> -u x = -u ( y + 1 ) )
14	13	oveq2d 6091	. . . . . . 7 |- ( x = ( y + 1 ) -> ( M + -u x ) = ( M + -u ( y + 1 ) ) )
15	14	eleq1d 2307	. . . . . 6 |- ( x = ( y + 1 ) -> ( ( M + -u x ) e. ZZ <-> ( M + -u ( y + 1 ) ) e. ZZ ) )
16	15	imbi2d 230	. . . . 5 |- ( x = ( y + 1 ) -> ( ( ( M e. ZZ /\ N e. RR ) -> ( M + -u x ) e. ZZ ) <-> ( ( M e. ZZ /\ N e. RR ) -> ( M + -u ( y + 1 ) ) e. ZZ ) ) )
17		negeq 8509	. . . . . . . 8 |- ( x = -u N -> -u x = -u -u N )
18	17	oveq2d 6091	. . . . . . 7 |- ( x = -u N -> ( M + -u x ) = ( M + -u -u N ) )
19	18	eleq1d 2307	. . . . . 6 |- ( x = -u N -> ( ( M + -u x ) e. ZZ <-> ( M + -u -u N ) e. ZZ ) )
20	19	imbi2d 230	. . . . 5 |- ( x = -u N -> ( ( ( M e. ZZ /\ N e. RR ) -> ( M + -u x ) e. ZZ ) <-> ( ( M e. ZZ /\ N e. RR ) -> ( M + -u -u N ) e. ZZ ) ) )
21		zcn 9628	. . . . . . . 8 |- ( M e. ZZ -> M e. CC )
22	21	adantr 276	. . . . . . 7 |- ( ( M e. ZZ /\ N e. RR ) -> M e. CC )
23		1cnd 8332	. . . . . . 7 |- ( ( M e. ZZ /\ N e. RR ) -> 1 e. CC )
24	22, 23	negsubd 8633	. . . . . 6 |- ( ( M e. ZZ /\ N e. RR ) -> ( M + -u 1 ) = ( M - 1 ) )
25		peano2zm 9661	. . . . . . 7 |- ( M e. ZZ -> ( M - 1 ) e. ZZ )
26	25	adantr 276	. . . . . 6 |- ( ( M e. ZZ /\ N e. RR ) -> ( M - 1 ) e. ZZ )
27	24, 26	eqeltrd 2315	. . . . 5 |- ( ( M e. ZZ /\ N e. RR ) -> ( M + -u 1 ) e. ZZ )
28		nncn 9291	. . . . . . . . . . 11 |- ( y e. NN -> y e. CC )
29	28	ad2antrr 492	. . . . . . . . . 10 |- ( ( ( y e. NN /\ ( M e. ZZ /\ N e. RR ) ) /\ ( M + -u y ) e. ZZ ) -> y e. CC )
30		1cnd 8332	. . . . . . . . . 10 |- ( ( ( y e. NN /\ ( M e. ZZ /\ N e. RR ) ) /\ ( M + -u y ) e. ZZ ) -> 1 e. CC )
31	29, 30	negdi2d 8641	. . . . . . . . 9 |- ( ( ( y e. NN /\ ( M e. ZZ /\ N e. RR ) ) /\ ( M + -u y ) e. ZZ ) -> -u ( y + 1 ) = ( -u y - 1 ) )
32	31	oveq2d 6091	. . . . . . . 8 |- ( ( ( y e. NN /\ ( M e. ZZ /\ N e. RR ) ) /\ ( M + -u y ) e. ZZ ) -> ( M + -u ( y + 1 ) ) = ( M + ( -u y - 1 ) ) )
33	22	ad2antlr 493	. . . . . . . . . 10 |- ( ( ( y e. NN /\ ( M e. ZZ /\ N e. RR ) ) /\ ( M + -u y ) e. ZZ ) -> M e. CC )
34	29	negcld 8614	. . . . . . . . . 10 |- ( ( ( y e. NN /\ ( M e. ZZ /\ N e. RR ) ) /\ ( M + -u y ) e. ZZ ) -> -u y e. CC )
35	33, 34, 30	addsubassd 8647	. . . . . . . . 9 |- ( ( ( y e. NN /\ ( M e. ZZ /\ N e. RR ) ) /\ ( M + -u y ) e. ZZ ) -> ( ( M + -u y ) - 1 ) = ( M + ( -u y - 1 ) ) )
36		peano2zm 9661	. . . . . . . . . 10 |- ( ( M + -u y ) e. ZZ -> ( ( M + -u y ) - 1 ) e. ZZ )
37	36	adantl 277	. . . . . . . . 9 |- ( ( ( y e. NN /\ ( M e. ZZ /\ N e. RR ) ) /\ ( M + -u y ) e. ZZ ) -> ( ( M + -u y ) - 1 ) e. ZZ )
38	35, 37	eqeltrrd 2316	. . . . . . . 8 |- ( ( ( y e. NN /\ ( M e. ZZ /\ N e. RR ) ) /\ ( M + -u y ) e. ZZ ) -> ( M + ( -u y - 1 ) ) e. ZZ )
39	32, 38	eqeltrd 2315	. . . . . . 7 |- ( ( ( y e. NN /\ ( M e. ZZ /\ N e. RR ) ) /\ ( M + -u y ) e. ZZ ) -> ( M + -u ( y + 1 ) ) e. ZZ )
40	39	exp31 364	. . . . . 6 |- ( y e. NN -> ( ( M e. ZZ /\ N e. RR ) -> ( ( M + -u y ) e. ZZ -> ( M + -u ( y + 1 ) ) e. ZZ ) ) )
41	40	a2d 26	. . . . 5 |- ( y e. NN -> ( ( ( M e. ZZ /\ N e. RR ) -> ( M + -u y ) e. ZZ ) -> ( ( M e. ZZ /\ N e. RR ) -> ( M + -u ( y + 1 ) ) e. ZZ ) ) )
42	8, 12, 16, 20, 27, 41	nnind 9299	. . . 4 |- ( -u N e. NN -> ( ( M e. ZZ /\ N e. RR ) -> ( M + -u -u N ) e. ZZ ) )
43	42	impcom 125	. . 3 |- ( ( ( M e. ZZ /\ N e. RR ) /\ -u N e. NN ) -> ( M + -u -u N ) e. ZZ )
44	43	3impa 1225	. 2 |- ( ( M e. ZZ /\ N e. RR /\ -u N e. NN ) -> ( M + -u -u N ) e. ZZ )
45	4, 44	eqeltrrd 2316	1 |- ( ( M e. ZZ /\ N e. RR /\ -u N e. NN ) -> ( M + N ) e. ZZ )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209  (class class class)co 6075   CCcc 8167   RRcr 8168   1c1 8170   + caddc 8172   - cmin 8487   -ucneg 8488   NNcn 9283   ZZcz 9623

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624

This theorem is referenced by:  zaddcl  9663