Theorem addnidpi 9723

Description: There is no identity element for addition on positive integers. (Contributed by NM, 28-Nov-1995.) (New usage is discouraged.)

Assertion

Ref	Expression
addnidpi	|- ( A e. N. -> -. ( A +N B ) = A )

Proof of Theorem addnidpi

Step	Hyp	Ref	Expression
1		pinn 9700	. . . . 5 |- ( A e. N. -> A e. om )
2		elni2 9699	. . . . . 6 |- ( B e. N. <-> ( B e. om /\ (/) e. B ) )
3		nnaordi 7698	. . . . . . . . 9 |- ( ( B e. om /\ A e. om ) -> ( (/) e. B -> ( A +o (/) ) e. ( A +o B ) ) )
4		nna0 7684	. . . . . . . . . . . 12 |- ( A e. om -> ( A +o (/) ) = A )
5	4	eleq1d 2686	. . . . . . . . . . 11 |- ( A e. om -> ( ( A +o (/) ) e. ( A +o B ) <-> A e. ( A +o B ) ) )
6		nnord 7073	. . . . . . . . . . . . . 14 |- ( A e. om -> Ord A )
7		ordirr 5741	. . . . . . . . . . . . . 14 |- ( Ord A -> -. A e. A )
8	6, 7	syl 17	. . . . . . . . . . . . 13 |- ( A e. om -> -. A e. A )
9		eleq2 2690	. . . . . . . . . . . . . 14 |- ( ( A +o B ) = A -> ( A e. ( A +o B ) <-> A e. A ) )
10	9	notbid 308	. . . . . . . . . . . . 13 |- ( ( A +o B ) = A -> ( -. A e. ( A +o B ) <-> -. A e. A ) )
11	8, 10	syl5ibrcom 237	. . . . . . . . . . . 12 |- ( A e. om -> ( ( A +o B ) = A -> -. A e. ( A +o B ) ) )
12	11	con2d 129	. . . . . . . . . . 11 |- ( A e. om -> ( A e. ( A +o B ) -> -. ( A +o B ) = A ) )
13	5, 12	sylbid 230	. . . . . . . . . 10 |- ( A e. om -> ( ( A +o (/) ) e. ( A +o B ) -> -. ( A +o B ) = A ) )
14	13	adantl 482	. . . . . . . . 9 |- ( ( B e. om /\ A e. om ) -> ( ( A +o (/) ) e. ( A +o B ) -> -. ( A +o B ) = A ) )
15	3, 14	syld 47	. . . . . . . 8 |- ( ( B e. om /\ A e. om ) -> ( (/) e. B -> -. ( A +o B ) = A ) )
16	15	expcom 451	. . . . . . 7 |- ( A e. om -> ( B e. om -> ( (/) e. B -> -. ( A +o B ) = A ) ) )
17	16	imp32 449	. . . . . 6 |- ( ( A e. om /\ ( B e. om /\ (/) e. B ) ) -> -. ( A +o B ) = A )
18	2, 17	sylan2b 492	. . . . 5 |- ( ( A e. om /\ B e. N. ) -> -. ( A +o B ) = A )
19	1, 18	sylan 488	. . . 4 |- ( ( A e. N. /\ B e. N. ) -> -. ( A +o B ) = A )
20		addpiord 9706	. . . . 5 |- ( ( A e. N. /\ B e. N. ) -> ( A +N B ) = ( A +o B ) )
21	20	eqeq1d 2624	. . . 4 |- ( ( A e. N. /\ B e. N. ) -> ( ( A +N B ) = A <-> ( A +o B ) = A ) )
22	19, 21	mtbird 315	. . 3 |- ( ( A e. N. /\ B e. N. ) -> -. ( A +N B ) = A )
23	22	a1d 25	. 2 |- ( ( A e. N. /\ B e. N. ) -> ( A e. N. -> -. ( A +N B ) = A ) )
24		dmaddpi 9712	. . . . . 6 |- dom +N = ( N. X. N. )
25	24	ndmov 6818	. . . . 5 |- ( -. ( A e. N. /\ B e. N. ) -> ( A +N B ) = (/) )
26	25	eqeq1d 2624	. . . 4 |- ( -. ( A e. N. /\ B e. N. ) -> ( ( A +N B ) = A <-> (/) = A ) )
27		0npi 9704	. . . . 5 |- -. (/) e. N.
28		eleq1 2689	. . . . 5 |- ( (/) = A -> ( (/) e. N. <-> A e. N. ) )
29	27, 28	mtbii 316	. . . 4 |- ( (/) = A -> -. A e. N. )
30	26, 29	syl6bi 243	. . 3 |- ( -. ( A e. N. /\ B e. N. ) -> ( ( A +N B ) = A -> -. A e. N. ) )
31	30	con2d 129	. 2 |- ( -. ( A e. N. /\ B e. N. ) -> ( A e. N. -> -. ( A +N B ) = A ) )
32	23, 31	pm2.61i 176	1 |- ( A e. N. -> -. ( A +N B ) = A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   (/)c0 3915   Ord word 5722  (class class class)co 6650   omcom 7065   +o coa 7557   N.cnpi 9666   +N cpli 9667

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

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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-ni 9694  df-pli 9695

This theorem is referenced by: (None)