Theorem zindd 9067

Description: Principle of Mathematical Induction on all integers, deduction version. The first five hypotheses give the substitutions; the last three are the basis, the induction, and the extension to negative numbers. (Contributed by Paul Chapman, 17-Apr-2009.) (Proof shortened by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
zindd.1	|- ( x = 0 -> ( ph <-> ps ) )
zindd.2	|- ( x = y -> ( ph <-> ch ) )
zindd.3	|- ( x = ( y + 1 ) -> ( ph <-> ta ) )
zindd.4	|- ( x = -u y -> ( ph <-> th ) )
zindd.5	|- ( x = A -> ( ph <-> et ) )
zindd.6	|- ( ze -> ps )
zindd.7	|- ( ze -> ( y e. NN0 -> ( ch -> ta ) ) )
zindd.8	|- ( ze -> ( y e. NN -> ( ch -> th ) ) )

Assertion

Ref	Expression
zindd	|- ( ze -> ( A e. ZZ -> et ) )

Distinct variable groups:   x, A   ch, x   et, x   ph, y   ps, x   ta, x   th, x   x, y, ze

Allowed substitution hints:   ph( x)   ps( y)   ch( y)   th( y)   ta( y)   et( y)   A( y)

Proof of Theorem zindd

Step	Hyp	Ref	Expression
1		znegcl 8983	. . . . . . 7 |- ( y e. ZZ -> -u y e. ZZ )
2		elznn0nn 8966	. . . . . . 7 |- ( -u y e. ZZ <-> ( -u y e. NN0 \/ ( -u y e. RR /\ -u -u y e. NN ) ) )
3	1, 2	sylib 121	. . . . . 6 |- ( y e. ZZ -> ( -u y e. NN0 \/ ( -u y e. RR /\ -u -u y e. NN ) ) )
4		simpr 109	. . . . . . 7 |- ( ( -u y e. RR /\ -u -u y e. NN ) -> -u -u y e. NN )
5	4	orim2i 733	. . . . . 6 |- ( ( -u y e. NN0 \/ ( -u y e. RR /\ -u -u y e. NN ) ) -> ( -u y e. NN0 \/ -u -u y e. NN ) )
6	3, 5	syl 14	. . . . 5 |- ( y e. ZZ -> ( -u y e. NN0 \/ -u -u y e. NN ) )
7		zcn 8957	. . . . . . . 8 |- ( y e. ZZ -> y e. CC )
8	7	negnegd 7981	. . . . . . 7 |- ( y e. ZZ -> -u -u y = y )
9	8	eleq1d 2181	. . . . . 6 |- ( y e. ZZ -> ( -u -u y e. NN <-> y e. NN ) )
10	9	orbi2d 762	. . . . 5 |- ( y e. ZZ -> ( ( -u y e. NN0 \/ -u -u y e. NN ) <-> ( -u y e. NN0 \/ y e. NN ) ) )
11	6, 10	mpbid 146	. . . 4 |- ( y e. ZZ -> ( -u y e. NN0 \/ y e. NN ) )
12		zindd.1	. . . . . . . 8 |- ( x = 0 -> ( ph <-> ps ) )
13	12	imbi2d 229	. . . . . . 7 |- ( x = 0 -> ( ( ze -> ph ) <-> ( ze -> ps ) ) )
14		zindd.2	. . . . . . . 8 |- ( x = y -> ( ph <-> ch ) )
15	14	imbi2d 229	. . . . . . 7 |- ( x = y -> ( ( ze -> ph ) <-> ( ze -> ch ) ) )
16		zindd.3	. . . . . . . 8 |- ( x = ( y + 1 ) -> ( ph <-> ta ) )
17	16	imbi2d 229	. . . . . . 7 |- ( x = ( y + 1 ) -> ( ( ze -> ph ) <-> ( ze -> ta ) ) )
18		zindd.4	. . . . . . . 8 |- ( x = -u y -> ( ph <-> th ) )
19	18	imbi2d 229	. . . . . . 7 |- ( x = -u y -> ( ( ze -> ph ) <-> ( ze -> th ) ) )
20		zindd.6	. . . . . . 7 |- ( ze -> ps )
21		zindd.7	. . . . . . . . 9 |- ( ze -> ( y e. NN0 -> ( ch -> ta ) ) )
22	21	com12 30	. . . . . . . 8 |- ( y e. NN0 -> ( ze -> ( ch -> ta ) ) )
23	22	a2d 26	. . . . . . 7 |- ( y e. NN0 -> ( ( ze -> ch ) -> ( ze -> ta ) ) )
24	13, 15, 17, 19, 20, 23	nn0ind 9063	. . . . . 6 |- ( -u y e. NN0 -> ( ze -> th ) )
25	24	com12 30	. . . . 5 |- ( ze -> ( -u y e. NN0 -> th ) )
26		nnnn0 8882	. . . . . . . 8 |- ( y e. NN -> y e. NN0 )
27	13, 15, 17, 15, 20, 23	nn0ind 9063	. . . . . . . 8 |- ( y e. NN0 -> ( ze -> ch ) )
28	26, 27	syl 14	. . . . . . 7 |- ( y e. NN -> ( ze -> ch ) )
29	28	com12 30	. . . . . 6 |- ( ze -> ( y e. NN -> ch ) )
30		zindd.8	. . . . . 6 |- ( ze -> ( y e. NN -> ( ch -> th ) ) )
31	29, 30	mpdd 41	. . . . 5 |- ( ze -> ( y e. NN -> th ) )
32	25, 31	jaod 689	. . . 4 |- ( ze -> ( ( -u y e. NN0 \/ y e. NN ) -> th ) )
33	11, 32	syl5 32	. . 3 |- ( ze -> ( y e. ZZ -> th ) )
34	33	ralrimiv 2476	. 2 |- ( ze -> A. y e. ZZ th )
35		znegcl 8983	. . . . 5 |- ( x e. ZZ -> -u x e. ZZ )
36		negeq 7872	. . . . . . . . 9 |- ( y = -u x -> -u y = -u -u x )
37		zcn 8957	. . . . . . . . . 10 |- ( x e. ZZ -> x e. CC )
38	37	negnegd 7981	. . . . . . . . 9 |- ( x e. ZZ -> -u -u x = x )
39	36, 38	sylan9eqr 2167	. . . . . . . 8 |- ( ( x e. ZZ /\ y = -u x ) -> -u y = x )
40	39	eqcomd 2118	. . . . . . 7 |- ( ( x e. ZZ /\ y = -u x ) -> x = -u y )
41	40, 18	syl 14	. . . . . 6 |- ( ( x e. ZZ /\ y = -u x ) -> ( ph <-> th ) )
42	41	bicomd 140	. . . . 5 |- ( ( x e. ZZ /\ y = -u x ) -> ( th <-> ph ) )
43	35, 42	rspcdv 2761	. . . 4 |- ( x e. ZZ -> ( A. y e. ZZ th -> ph ) )
44	43	com12 30	. . 3 |- ( A. y e. ZZ th -> ( x e. ZZ -> ph ) )
45	44	ralrimiv 2476	. 2 |- ( A. y e. ZZ th -> A. x e. ZZ ph )
46		zindd.5	. . 3 |- ( x = A -> ( ph <-> et ) )
47	46	rspccv 2755	. 2 |- ( A. x e. ZZ ph -> ( A e. ZZ -> et ) )
48	34, 45, 47	3syl 17	1 |- ( ze -> ( A e. ZZ -> et ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 680   = wceq 1312   e. wcel 1461   A.wral 2388  (class class class)co 5726   RRcr 7540   0cc0 7541   1c1 7542   + caddc 7544   -ucneg 7851   NNcn 8624   NN0cn0 8875   ZZcz 8952

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-cnex 7630  ax-resscn 7631  ax-1cn 7632  ax-1re 7633  ax-icn 7634  ax-addcl 7635  ax-addrcl 7636  ax-mulcl 7637  ax-addcom 7639  ax-addass 7641  ax-distr 7643  ax-i2m1 7644  ax-0lt1 7645  ax-0id 7647  ax-rnegex 7648  ax-cnre 7650  ax-pre-ltirr 7651  ax-pre-ltwlin 7652  ax-pre-lttrn 7653  ax-pre-ltadd 7655

This theorem depends on definitions:  df-bi 116  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-br 3894  df-opab 3948  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-iota 5044  df-fun 5081  df-fv 5087  df-riota 5682  df-ov 5729  df-oprab 5730  df-mpo 5731  df-pnf 7720  df-mnf 7721  df-xr 7722  df-ltxr 7723  df-le 7724  df-sub 7852  df-neg 7853  df-inn 8625  df-n0 8876  df-z 8953

This theorem is referenced by:  efexp  11233