Theorem preaddccan2 4456

Description: Cancellation law for natural addition with a non-null condition. (Contributed by SF, 29-Jan-2015.)

Assertion

Ref	Expression
preaddccan2	|- (((M e. Nn /\ N e. Nn /\ P e. Nn ) /\ (M +o N) =/= (/)) -> ((M +o N) = (M +o P) <-> N = P))

Proof of Theorem preaddccan2

Dummy variables m k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		preaddccan2lem1 4455	. . . . 5 |- ((N e. Nn /\ P e. Nn ) -> {m | (((m +o N) =/= (/) /\ (m +o N) = (m +o P)) -> N = P)} e. _V)
2		addceq1 4384	. . . . . . . 8 |- (m = 0c -> (m +o N) = (0c +o N))
3	2	neeq1d 2530	. . . . . . 7 |- (m = 0c -> ((m +o N) =/= (/) <-> (0c +o N) =/= (/)))
4		addceq1 4384	. . . . . . . 8 |- (m = 0c -> (m +o P) = (0c +o P))
5	2, 4	eqeq12d 2367	. . . . . . 7 |- (m = 0c -> ((m +o N) = (m +o P) <-> (0c +o N) = (0c +o P)))
6	3, 5	anbi12d 691	. . . . . 6 |- (m = 0c -> (((m +o N) =/= (/) /\ (m +o N) = (m +o P)) <-> ((0c +o N) =/= (/) /\ (0c +o N) = (0c +o P))))
7	6	imbi1d 308	. . . . 5 |- (m = 0c -> ((((m +o N) =/= (/) /\ (m +o N) = (m +o P)) -> N = P) <-> (((0c +o N) =/= (/) /\ (0c +o N) = (0c +o P)) -> N = P)))
8		addceq1 4384	. . . . . . . 8 |- (m = k -> (m +o N) = (k +o N))
9	8	neeq1d 2530	. . . . . . 7 |- (m = k -> ((m +o N) =/= (/) <-> (k +o N) =/= (/)))
10		addceq1 4384	. . . . . . . 8 |- (m = k -> (m +o P) = (k +o P))
11	8, 10	eqeq12d 2367	. . . . . . 7 |- (m = k -> ((m +o N) = (m +o P) <-> (k +o N) = (k +o P)))
12	9, 11	anbi12d 691	. . . . . 6 |- (m = k -> (((m +o N) =/= (/) /\ (m +o N) = (m +o P)) <-> ((k +o N) =/= (/) /\ (k +o N) = (k +o P))))
13	12	imbi1d 308	. . . . 5 |- (m = k -> ((((m +o N) =/= (/) /\ (m +o N) = (m +o P)) -> N = P) <-> (((k +o N) =/= (/) /\ (k +o N) = (k +o P)) -> N = P)))
14		addceq1 4384	. . . . . . . . 9 |- (m = (k +o 1c) -> (m +o N) = ((k +o 1c) +o N))
15		addc32 4417	. . . . . . . . 9 |- ((k +o 1c) +o N) = ((k +o N) +o 1c)
16	14, 15	syl6eq 2401	. . . . . . . 8 |- (m = (k +o 1c) -> (m +o N) = ((k +o N) +o 1c))
17	16	neeq1d 2530	. . . . . . 7 |- (m = (k +o 1c) -> ((m +o N) =/= (/) <-> ((k +o N) +o 1c) =/= (/)))
18		addceq1 4384	. . . . . . . . 9 |- (m = (k +o 1c) -> (m +o P) = ((k +o 1c) +o P))
19		addc32 4417	. . . . . . . . 9 |- ((k +o 1c) +o P) = ((k +o P) +o 1c)
20	18, 19	syl6eq 2401	. . . . . . . 8 |- (m = (k +o 1c) -> (m +o P) = ((k +o P) +o 1c))
21	16, 20	eqeq12d 2367	. . . . . . 7 |- (m = (k +o 1c) -> ((m +o N) = (m +o P) <-> ((k +o N) +o 1c) = ((k +o P) +o 1c)))
22	17, 21	anbi12d 691	. . . . . 6 |- (m = (k +o 1c) -> (((m +o N) =/= (/) /\ (m +o N) = (m +o P)) <-> (((k +o N) +o 1c) =/= (/) /\ ((k +o N) +o 1c) = ((k +o P) +o 1c))))
23	22	imbi1d 308	. . . . 5 |- (m = (k +o 1c) -> ((((m +o N) =/= (/) /\ (m +o N) = (m +o P)) -> N = P) <-> ((((k +o N) +o 1c) =/= (/) /\ ((k +o N) +o 1c) = ((k +o P) +o 1c)) -> N = P)))
24		addceq1 4384	. . . . . . . 8 |- (m = M -> (m +o N) = (M +o N))
25	24	neeq1d 2530	. . . . . . 7 |- (m = M -> ((m +o N) =/= (/) <-> (M +o N) =/= (/)))
26		addceq1 4384	. . . . . . . 8 |- (m = M -> (m +o P) = (M +o P))
27	24, 26	eqeq12d 2367	. . . . . . 7 |- (m = M -> ((m +o N) = (m +o P) <-> (M +o N) = (M +o P)))
28	25, 27	anbi12d 691	. . . . . 6 |- (m = M -> (((m +o N) =/= (/) /\ (m +o N) = (m +o P)) <-> ((M +o N) =/= (/) /\ (M +o N) = (M +o P))))
29	28	imbi1d 308	. . . . 5 |- (m = M -> ((((m +o N) =/= (/) /\ (m +o N) = (m +o P)) -> N = P) <-> (((M +o N) =/= (/) /\ (M +o N) = (M +o P)) -> N = P)))
30		addcid2 4408	. . . . . . . . 9 |- (0c +o N) = N
31		addcid2 4408	. . . . . . . . 9 |- (0c +o P) = P
32	30, 31	eqeq12i 2366	. . . . . . . 8 |- ((0c +o N) = (0c +o P) <-> N = P)
33	32	biimpi 186	. . . . . . 7 |- ((0c +o N) = (0c +o P) -> N = P)
34	33	adantl 452	. . . . . 6 |- (((0c +o N) =/= (/) /\ (0c +o N) = (0c +o P)) -> N = P)
35	34	a1i 10	. . . . 5 |- ((N e. Nn /\ P e. Nn ) -> (((0c +o N) =/= (/) /\ (0c +o N) = (0c +o P)) -> N = P))
36		addcnnul 4454	. . . . . . . . . 10 |- (((k +o N) +o 1c) =/= (/) -> ((k +o N) =/= (/) /\ 1c =/= (/)))
37	36	simpld 445	. . . . . . . . 9 |- (((k +o N) +o 1c) =/= (/) -> (k +o N) =/= (/))
38	37	ad2antrl 708	. . . . . . . 8 |- (((k e. Nn /\ (N e. Nn /\ P e. Nn )) /\ (((k +o N) +o 1c) =/= (/) /\ ((k +o N) +o 1c) = ((k +o P) +o 1c))) -> (k +o N) =/= (/))
39		simpll 730	. . . . . . . . . 10 |- (((k e. Nn /\ (N e. Nn /\ P e. Nn )) /\ (((k +o N) +o 1c) =/= (/) /\ ((k +o N) +o 1c) = ((k +o P) +o 1c))) -> k e. Nn )
40		simplrl 736	. . . . . . . . . 10 |- (((k e. Nn /\ (N e. Nn /\ P e. Nn )) /\ (((k +o N) +o 1c) =/= (/) /\ ((k +o N) +o 1c) = ((k +o P) +o 1c))) -> N e. Nn )
41		nncaddccl 4420	. . . . . . . . . 10 |- ((k e. Nn /\ N e. Nn ) -> (k +o N) e. Nn )
42	39, 40, 41	syl2anc 642	. . . . . . . . 9 |- (((k e. Nn /\ (N e. Nn /\ P e. Nn )) /\ (((k +o N) +o 1c) =/= (/) /\ ((k +o N) +o 1c) = ((k +o P) +o 1c))) -> (k +o N) e. Nn )
43		simplrr 737	. . . . . . . . . 10 |- (((k e. Nn /\ (N e. Nn /\ P e. Nn )) /\ (((k +o N) +o 1c) =/= (/) /\ ((k +o N) +o 1c) = ((k +o P) +o 1c))) -> P e. Nn )
44		nncaddccl 4420	. . . . . . . . . 10 |- ((k e. Nn /\ P e. Nn ) -> (k +o P) e. Nn )
45	39, 43, 44	syl2anc 642	. . . . . . . . 9 |- (((k e. Nn /\ (N e. Nn /\ P e. Nn )) /\ (((k +o N) +o 1c) =/= (/) /\ ((k +o N) +o 1c) = ((k +o P) +o 1c))) -> (k +o P) e. Nn )
46		simprr 733	. . . . . . . . 9 |- (((k e. Nn /\ (N e. Nn /\ P e. Nn )) /\ (((k +o N) +o 1c) =/= (/) /\ ((k +o N) +o 1c) = ((k +o P) +o 1c))) -> ((k +o N) +o 1c) = ((k +o P) +o 1c))
47		simprl 732	. . . . . . . . 9 |- (((k e. Nn /\ (N e. Nn /\ P e. Nn )) /\ (((k +o N) +o 1c) =/= (/) /\ ((k +o N) +o 1c) = ((k +o P) +o 1c))) -> ((k +o N) +o 1c) =/= (/))
48		prepeano4 4452	. . . . . . . . 9 |- ((((k +o N) e. Nn /\ (k +o P) e. Nn ) /\ (((k +o N) +o 1c) = ((k +o P) +o 1c) /\ ((k +o N) +o 1c) =/= (/))) -> (k +o N) = (k +o P))
49	42, 45, 46, 47, 48	syl22anc 1183	. . . . . . . 8 |- (((k e. Nn /\ (N e. Nn /\ P e. Nn )) /\ (((k +o N) +o 1c) =/= (/) /\ ((k +o N) +o 1c) = ((k +o P) +o 1c))) -> (k +o N) = (k +o P))
50	38, 49	jca 518	. . . . . . 7 |- (((k e. Nn /\ (N e. Nn /\ P e. Nn )) /\ (((k +o N) +o 1c) =/= (/) /\ ((k +o N) +o 1c) = ((k +o P) +o 1c))) -> ((k +o N) =/= (/) /\ (k +o N) = (k +o P)))
51	50	ex 423	. . . . . 6 |- ((k e. Nn /\ (N e. Nn /\ P e. Nn )) -> ((((k +o N) +o 1c) =/= (/) /\ ((k +o N) +o 1c) = ((k +o P) +o 1c)) -> ((k +o N) =/= (/) /\ (k +o N) = (k +o P))))
52	51	imim1d 69	. . . . 5 |- ((k e. Nn /\ (N e. Nn /\ P e. Nn )) -> ((((k +o N) =/= (/) /\ (k +o N) = (k +o P)) -> N = P) -> ((((k +o N) +o 1c) =/= (/) /\ ((k +o N) +o 1c) = ((k +o P) +o 1c)) -> N = P)))
53	1, 7, 13, 23, 29, 35, 52	findsd 4411	. . . 4 |- ((M e. Nn /\ (N e. Nn /\ P e. Nn )) -> (((M +o N) =/= (/) /\ (M +o N) = (M +o P)) -> N = P))
54	53	3impb 1147	. . 3 |- ((M e. Nn /\ N e. Nn /\ P e. Nn ) -> (((M +o N) =/= (/) /\ (M +o N) = (M +o P)) -> N = P))
55	54	expdimp 426	. 2 |- (((M e. Nn /\ N e. Nn /\ P e. Nn ) /\ (M +o N) =/= (/)) -> ((M +o N) = (M +o P) -> N = P))
56		addceq2 4385	. 2 |- (N = P -> (M +o N) = (M +o P))
57	55, 56	impbid1 194	1 |- (((M e. Nn /\ N e. Nn /\ P e. Nn ) /\ (M +o N) =/= (/)) -> ((M +o N) = (M +o P) <-> N = P))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934   = wceq 1642   e. wcel 1710   =/= wne 2517  _Vcvv 2860  (/)c0 3551  1_cc1c 4135   Nn cnnc 4374  0_cc0c 4375   +o cplc 4376

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-0c 4378  df-addc 4379  df-nnc 4380

This theorem is referenced by:  ltfinirr  4458  vfin1cltv  4548  addccan2  4560