Theorem zindd 9526

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 9438	. . . . . . 7 |- ( y e. ZZ -> -u y e. ZZ )
2		elznn0nn 9421	. . . . . . 7 |- ( -u y e. ZZ <-> ( -u y e. NN0 \/ ( -u y e. RR /\ -u -u y e. NN ) ) )
3	1, 2	sylib 122	. . . . . 6 |- ( y e. ZZ -> ( -u y e. NN0 \/ ( -u y e. RR /\ -u -u y e. NN ) ) )
4		simpr 110	. . . . . . 7 |- ( ( -u y e. RR /\ -u -u y e. NN ) -> -u -u y e. NN )
5	4	orim2i 763	. . . . . 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 9412	. . . . . . . 8 |- ( y e. ZZ -> y e. CC )
8	7	negnegd 8409	. . . . . . 7 |- ( y e. ZZ -> -u -u y = y )
9	8	eleq1d 2276	. . . . . 6 |- ( y e. ZZ -> ( -u -u y e. NN <-> y e. NN ) )
10	9	orbi2d 792	. . . . 5 |- ( y e. ZZ -> ( ( -u y e. NN0 \/ -u -u y e. NN ) <-> ( -u y e. NN0 \/ y e. NN ) ) )
11	6, 10	mpbid 147	. . . 4 |- ( y e. ZZ -> ( -u y e. NN0 \/ y e. NN ) )
12		zindd.1	. . . . . . . 8 |- ( x = 0 -> ( ph <-> ps ) )
13	12	imbi2d 230	. . . . . . 7 |- ( x = 0 -> ( ( ze -> ph ) <-> ( ze -> ps ) ) )
14		zindd.2	. . . . . . . 8 |- ( x = y -> ( ph <-> ch ) )
15	14	imbi2d 230	. . . . . . 7 |- ( x = y -> ( ( ze -> ph ) <-> ( ze -> ch ) ) )
16		zindd.3	. . . . . . . 8 |- ( x = ( y + 1 ) -> ( ph <-> ta ) )
17	16	imbi2d 230	. . . . . . 7 |- ( x = ( y + 1 ) -> ( ( ze -> ph ) <-> ( ze -> ta ) ) )
18		zindd.4	. . . . . . . 8 |- ( x = -u y -> ( ph <-> th ) )
19	18	imbi2d 230	. . . . . . 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 9522	. . . . . 6 |- ( -u y e. NN0 -> ( ze -> th ) )
25	24	com12 30	. . . . 5 |- ( ze -> ( -u y e. NN0 -> th ) )
26		nnnn0 9337	. . . . . . . 8 |- ( y e. NN -> y e. NN0 )
27	13, 15, 17, 15, 20, 23	nn0ind 9522	. . . . . . . 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 719	. . . 4 |- ( ze -> ( ( -u y e. NN0 \/ y e. NN ) -> th ) )
33	11, 32	syl5 32	. . 3 |- ( ze -> ( y e. ZZ -> th ) )
34	33	ralrimiv 2580	. 2 |- ( ze -> A. y e. ZZ th )
35		znegcl 9438	. . . . 5 |- ( x e. ZZ -> -u x e. ZZ )
36		negeq 8300	. . . . . . . . 9 |- ( y = -u x -> -u y = -u -u x )
37		zcn 9412	. . . . . . . . . 10 |- ( x e. ZZ -> x e. CC )
38	37	negnegd 8409	. . . . . . . . 9 |- ( x e. ZZ -> -u -u x = x )
39	36, 38	sylan9eqr 2262	. . . . . . . 8 |- ( ( x e. ZZ /\ y = -u x ) -> -u y = x )
40	39	eqcomd 2213	. . . . . . 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 141	. . . . 5 |- ( ( x e. ZZ /\ y = -u x ) -> ( th <-> ph ) )
43	35, 42	rspcdv 2887	. . . 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 2580	. 2 |- ( A. y e. ZZ th -> A. x e. ZZ ph )
46		zindd.5	. . 3 |- ( x = A -> ( ph <-> et ) )
47	46	rspccv 2881	. 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 104   <-> wb 105   \/ wo 710   = wceq 1373   e. wcel 2178   A.wral 2486  (class class class)co 5967   RRcr 7959   0cc0 7960   1c1 7961   + caddc 7963   -ucneg 8279   NNcn 9071   NN0cn0 9330   ZZcz 9407

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-inn 9072  df-n0 9331  df-z 9408

This theorem is referenced by:  efexp  12108  pcexp  12747  mulgaddcom  13597  mulginvcom  13598  mulgneg2  13607  mulgass2  13935  cnfldmulg  14453