Theorem zindd 9330

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 9243	. . . . . . 7 |- ( y e. ZZ -> -u y e. ZZ )
2		elznn0nn 9226	. . . . . . 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 756	. . . . . 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 9217	. . . . . . . 8 |- ( y e. ZZ -> y e. CC )
8	7	negnegd 8221	. . . . . . 7 |- ( y e. ZZ -> -u -u y = y )
9	8	eleq1d 2239	. . . . . 6 |- ( y e. ZZ -> ( -u -u y e. NN <-> y e. NN ) )
10	9	orbi2d 785	. . . . 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 9326	. . . . . 6 |- ( -u y e. NN0 -> ( ze -> th ) )
25	24	com12 30	. . . . 5 |- ( ze -> ( -u y e. NN0 -> th ) )
26		nnnn0 9142	. . . . . . . 8 |- ( y e. NN -> y e. NN0 )
27	13, 15, 17, 15, 20, 23	nn0ind 9326	. . . . . . . 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 712	. . . 4 |- ( ze -> ( ( -u y e. NN0 \/ y e. NN ) -> th ) )
33	11, 32	syl5 32	. . 3 |- ( ze -> ( y e. ZZ -> th ) )
34	33	ralrimiv 2542	. 2 |- ( ze -> A. y e. ZZ th )
35		znegcl 9243	. . . . 5 |- ( x e. ZZ -> -u x e. ZZ )
36		negeq 8112	. . . . . . . . 9 |- ( y = -u x -> -u y = -u -u x )
37		zcn 9217	. . . . . . . . . 10 |- ( x e. ZZ -> x e. CC )
38	37	negnegd 8221	. . . . . . . . 9 |- ( x e. ZZ -> -u -u x = x )
39	36, 38	sylan9eqr 2225	. . . . . . . 8 |- ( ( x e. ZZ /\ y = -u x ) -> -u y = x )
40	39	eqcomd 2176	. . . . . . 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 2837	. . . 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 2542	. 2 |- ( A. y e. ZZ th -> A. x e. ZZ ph )
46		zindd.5	. . 3 |- ( x = A -> ( ph <-> et ) )
47	46	rspccv 2831	. 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 703   = wceq 1348   e. wcel 2141   A.wral 2448  (class class class)co 5853   RRcr 7773   0cc0 7774   1c1 7775   + caddc 7777   -ucneg 8091   NNcn 8878   NN0cn0 9135   ZZcz 9212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213

This theorem is referenced by:  efexp  11645  pcexp  12263