Theorem fvmptnn04if 20654

Description: The function values of a mapping from the nonnegative integers with four distinct cases. (Contributed by AV, 10-Nov-2019.)

Hypotheses

Ref	Expression
fvmptnn04if.g	|- G = ( n e. NN0 |-> if ( n = 0 , A , if ( n = S , C , if ( S < n , D , B ) ) ) )
fvmptnn04if.s	|- ( ph -> S e. NN )
fvmptnn04if.n	|- ( ph -> N e. NN0 )
fvmptnn04if.y	|- ( ph -> Y e. V )
fvmptnn04if.a	|- ( ( ph /\ N = 0 ) -> Y = [_ N / n ]_ A )
fvmptnn04if.b	|- ( ( ph /\ 0 < N /\ N < S ) -> Y = [_ N / n ]_ B )
fvmptnn04if.c	|- ( ( ph /\ N = S ) -> Y = [_ N / n ]_ C )
fvmptnn04if.d	|- ( ( ph /\ S < N ) -> Y = [_ N / n ]_ D )

Assertion

Ref	Expression
fvmptnn04if	|- ( ph -> ( G ` N ) = Y )

Distinct variable groups:   n, N   S, n

Allowed substitution hints:   ph( n)   A( n)   B( n)   C( n)   D( n)   G( n)   V( n)   Y( n)

Proof of Theorem fvmptnn04if

Step	Hyp	Ref	Expression
1		fvmptnn04if.n	. . 3 |- ( ph -> N e. NN0 )
2		csbif 4138	. . . . 5 |- [_ N / n ]_ if ( n = 0 , A , if ( n = S , C , if ( S < n , D , B ) ) ) = if ( [. N / n ]. n = 0 , [_ N / n ]_ A , [_ N / n ]_ if ( n = S , C , if ( S < n , D , B ) ) )
3		eqsbc3 3475	. . . . . . 7 |- ( N e. NN0 -> ( [. N / n ]. n = 0 <-> N = 0 ) )
4	1, 3	syl 17	. . . . . 6 |- ( ph -> ( [. N / n ]. n = 0 <-> N = 0 ) )
5		csbif 4138	. . . . . . 7 |- [_ N / n ]_ if ( n = S , C , if ( S < n , D , B ) ) = if ( [. N / n ]. n = S , [_ N / n ]_ C , [_ N / n ]_ if ( S < n , D , B ) )
6		eqsbc3 3475	. . . . . . . . 9 |- ( N e. NN0 -> ( [. N / n ]. n = S <-> N = S ) )
7	1, 6	syl 17	. . . . . . . 8 |- ( ph -> ( [. N / n ]. n = S <-> N = S ) )
8		csbif 4138	. . . . . . . . 9 |- [_ N / n ]_ if ( S < n , D , B ) = if ( [. N / n ]. S < n , [_ N / n ]_ D , [_ N / n ]_ B )
9		sbcbr2g 4710	. . . . . . . . . . . 12 |- ( N e. NN0 -> ( [. N / n ]. S < n <-> S < [_ N / n ]_ n ) )
10	1, 9	syl 17	. . . . . . . . . . 11 |- ( ph -> ( [. N / n ]. S < n <-> S < [_ N / n ]_ n ) )
11		csbvarg 4003	. . . . . . . . . . . . 13 |- ( N e. NN0 -> [_ N / n ]_ n = N )
12	1, 11	syl 17	. . . . . . . . . . . 12 |- ( ph -> [_ N / n ]_ n = N )
13	12	breq2d 4665	. . . . . . . . . . 11 |- ( ph -> ( S < [_ N / n ]_ n <-> S < N ) )
14	10, 13	bitrd 268	. . . . . . . . . 10 |- ( ph -> ( [. N / n ]. S < n <-> S < N ) )
15	14	ifbid 4108	. . . . . . . . 9 |- ( ph -> if ( [. N / n ]. S < n , [_ N / n ]_ D , [_ N / n ]_ B ) = if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) )
16	8, 15	syl5eq 2668	. . . . . . . 8 |- ( ph -> [_ N / n ]_ if ( S < n , D , B ) = if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) )
17	7, 16	ifbieq2d 4111	. . . . . . 7 |- ( ph -> if ( [. N / n ]. n = S , [_ N / n ]_ C , [_ N / n ]_ if ( S < n , D , B ) ) = if ( N = S , [_ N / n ]_ C , if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) ) )
18	5, 17	syl5eq 2668	. . . . . 6 |- ( ph -> [_ N / n ]_ if ( n = S , C , if ( S < n , D , B ) ) = if ( N = S , [_ N / n ]_ C , if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) ) )
19	4, 18	ifbieq2d 4111	. . . . 5 |- ( ph -> if ( [. N / n ]. n = 0 , [_ N / n ]_ A , [_ N / n ]_ if ( n = S , C , if ( S < n , D , B ) ) ) = if ( N = 0 , [_ N / n ]_ A , if ( N = S , [_ N / n ]_ C , if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) ) ) )
20	2, 19	syl5eq 2668	. . . 4 |- ( ph -> [_ N / n ]_ if ( n = 0 , A , if ( n = S , C , if ( S < n , D , B ) ) ) = if ( N = 0 , [_ N / n ]_ A , if ( N = S , [_ N / n ]_ C , if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) ) ) )
21		fvmptnn04if.a	. . . . . 6 |- ( ( ph /\ N = 0 ) -> Y = [_ N / n ]_ A )
22		fvmptnn04if.y	. . . . . . 7 |- ( ph -> Y e. V )
23	22	adantr 481	. . . . . 6 |- ( ( ph /\ N = 0 ) -> Y e. V )
24	21, 23	eqeltrrd 2702	. . . . 5 |- ( ( ph /\ N = 0 ) -> [_ N / n ]_ A e. V )
25		fvmptnn04if.c	. . . . . . . . 9 |- ( ( ph /\ N = S ) -> Y = [_ N / n ]_ C )
26	25	eqcomd 2628	. . . . . . . 8 |- ( ( ph /\ N = S ) -> [_ N / n ]_ C = Y )
27	26	adantlr 751	. . . . . . 7 |- ( ( ( ph /\ -. N = 0 ) /\ N = S ) -> [_ N / n ]_ C = Y )
28	22	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ -. N = 0 ) /\ N = S ) -> Y e. V )
29	27, 28	eqeltrd 2701	. . . . . 6 |- ( ( ( ph /\ -. N = 0 ) /\ N = S ) -> [_ N / n ]_ C e. V )
30		fvmptnn04if.d	. . . . . . . . . . . 12 |- ( ( ph /\ S < N ) -> Y = [_ N / n ]_ D )
31	30	eqcomd 2628	. . . . . . . . . . 11 |- ( ( ph /\ S < N ) -> [_ N / n ]_ D = Y )
32	31	ex 450	. . . . . . . . . 10 |- ( ph -> ( S < N -> [_ N / n ]_ D = Y ) )
33	32	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) -> ( S < N -> [_ N / n ]_ D = Y ) )
34	33	imp 445	. . . . . . . 8 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ S < N ) -> [_ N / n ]_ D = Y )
35	22	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ S < N ) -> Y e. V )
36	34, 35	eqeltrd 2701	. . . . . . 7 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ S < N ) -> [_ N / n ]_ D e. V )
37		simplll 798	. . . . . . . . 9 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) -> ph )
38		ancom 466	. . . . . . . . . . . . 13 |- ( ( -. S < N /\ ph ) <-> ( ph /\ -. S < N ) )
39	38	anbi2i 730	. . . . . . . . . . . 12 |- ( ( ( -. N = 0 /\ -. N = S ) /\ ( -. S < N /\ ph ) ) <-> ( ( -. N = 0 /\ -. N = S ) /\ ( ph /\ -. S < N ) ) )
40		ancom 466	. . . . . . . . . . . . 13 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) <-> ( ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) /\ ph ) )
41		anass 681	. . . . . . . . . . . . . . 15 |- ( ( ( -. N = 0 /\ -. N = S ) /\ -. S < N ) <-> ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) )
42	41	bicomi 214	. . . . . . . . . . . . . 14 |- ( ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) <-> ( ( -. N = 0 /\ -. N = S ) /\ -. S < N ) )
43	42	anbi1i 731	. . . . . . . . . . . . 13 |- ( ( ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) /\ ph ) <-> ( ( ( -. N = 0 /\ -. N = S ) /\ -. S < N ) /\ ph ) )
44		anass 681	. . . . . . . . . . . . 13 |- ( ( ( ( -. N = 0 /\ -. N = S ) /\ -. S < N ) /\ ph ) <-> ( ( -. N = 0 /\ -. N = S ) /\ ( -. S < N /\ ph ) ) )
45	40, 43, 44	3bitri 286	. . . . . . . . . . . 12 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) <-> ( ( -. N = 0 /\ -. N = S ) /\ ( -. S < N /\ ph ) ) )
46		anass 681	. . . . . . . . . . . 12 |- ( ( ( ( -. N = 0 /\ -. N = S ) /\ ph ) /\ -. S < N ) <-> ( ( -. N = 0 /\ -. N = S ) /\ ( ph /\ -. S < N ) ) )
47	39, 45, 46	3bitr4i 292	. . . . . . . . . . 11 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) <-> ( ( ( -. N = 0 /\ -. N = S ) /\ ph ) /\ -. S < N ) )
48		an32 839	. . . . . . . . . . . . 13 |- ( ( ( -. N = 0 /\ -. N = S ) /\ ph ) <-> ( ( -. N = 0 /\ ph ) /\ -. N = S ) )
49		ancom 466	. . . . . . . . . . . . . 14 |- ( ( -. N = 0 /\ ph ) <-> ( ph /\ -. N = 0 ) )
50	49	anbi1i 731	. . . . . . . . . . . . 13 |- ( ( ( -. N = 0 /\ ph ) /\ -. N = S ) <-> ( ( ph /\ -. N = 0 ) /\ -. N = S ) )
51	48, 50	bitri 264	. . . . . . . . . . . 12 |- ( ( ( -. N = 0 /\ -. N = S ) /\ ph ) <-> ( ( ph /\ -. N = 0 ) /\ -. N = S ) )
52	51	anbi1i 731	. . . . . . . . . . 11 |- ( ( ( ( -. N = 0 /\ -. N = S ) /\ ph ) /\ -. S < N ) <-> ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) )
53	47, 52	bitri 264	. . . . . . . . . 10 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) <-> ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) )
54		df-ne 2795	. . . . . . . . . . . . 13 |- ( N =/= 0 <-> -. N = 0 )
55		elnnne0 11306	. . . . . . . . . . . . . . 15 |- ( N e. NN <-> ( N e. NN0 /\ N =/= 0 ) )
56		nngt0 11049	. . . . . . . . . . . . . . 15 |- ( N e. NN -> 0 < N )
57	55, 56	sylbir 225	. . . . . . . . . . . . . 14 |- ( ( N e. NN0 /\ N =/= 0 ) -> 0 < N )
58	57	expcom 451	. . . . . . . . . . . . 13 |- ( N =/= 0 -> ( N e. NN0 -> 0 < N ) )
59	54, 58	sylbir 225	. . . . . . . . . . . 12 |- ( -. N = 0 -> ( N e. NN0 -> 0 < N ) )
60	59	adantr 481	. . . . . . . . . . 11 |- ( ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) -> ( N e. NN0 -> 0 < N ) )
61	1, 60	mpan9 486	. . . . . . . . . 10 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> 0 < N )
62	53, 61	sylbir 225	. . . . . . . . 9 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) -> 0 < N )
63	1	nn0red 11352	. . . . . . . . . . . . . . . 16 |- ( ph -> N e. RR )
64		fvmptnn04if.s	. . . . . . . . . . . . . . . . 17 |- ( ph -> S e. NN )
65	64	nnred 11035	. . . . . . . . . . . . . . . 16 |- ( ph -> S e. RR )
66	63, 65	lenltd 10183	. . . . . . . . . . . . . . 15 |- ( ph -> ( N <_ S <-> -. S < N ) )
67	66	biimprd 238	. . . . . . . . . . . . . 14 |- ( ph -> ( -. S < N -> N <_ S ) )
68	67	adantld 483	. . . . . . . . . . . . 13 |- ( ph -> ( ( -. N = S /\ -. S < N ) -> N <_ S ) )
69	68	adantld 483	. . . . . . . . . . . 12 |- ( ph -> ( ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) -> N <_ S ) )
70	69	imp 445	. . . . . . . . . . 11 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> N <_ S )
71		nesym 2850	. . . . . . . . . . . . . 14 |- ( S =/= N <-> -. N = S )
72	71	biimpri 218	. . . . . . . . . . . . 13 |- ( -. N = S -> S =/= N )
73	72	adantr 481	. . . . . . . . . . . 12 |- ( ( -. N = S /\ -. S < N ) -> S =/= N )
74	73	ad2antll 765	. . . . . . . . . . 11 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> S =/= N )
75	63	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> N e. RR )
76	65	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> S e. RR )
77	75, 76	ltlend 10182	. . . . . . . . . . 11 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> ( N < S <-> ( N <_ S /\ S =/= N ) ) )
78	70, 74, 77	mpbir2and 957	. . . . . . . . . 10 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> N < S )
79	53, 78	sylbir 225	. . . . . . . . 9 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) -> N < S )
80		fvmptnn04if.b	. . . . . . . . . 10 |- ( ( ph /\ 0 < N /\ N < S ) -> Y = [_ N / n ]_ B )
81	80	eqcomd 2628	. . . . . . . . 9 |- ( ( ph /\ 0 < N /\ N < S ) -> [_ N / n ]_ B = Y )
82	37, 62, 79, 81	syl3anc 1326	. . . . . . . 8 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) -> [_ N / n ]_ B = Y )
83	22	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) -> Y e. V )
84	82, 83	eqeltrd 2701	. . . . . . 7 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) -> [_ N / n ]_ B e. V )
85	36, 84	ifclda 4120	. . . . . 6 |- ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) -> if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) e. V )
86	29, 85	ifclda 4120	. . . . 5 |- ( ( ph /\ -. N = 0 ) -> if ( N = S , [_ N / n ]_ C , if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) ) e. V )
87	24, 86	ifclda 4120	. . . 4 |- ( ph -> if ( N = 0 , [_ N / n ]_ A , if ( N = S , [_ N / n ]_ C , if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) ) ) e. V )
88	20, 87	eqeltrd 2701	. . 3 |- ( ph -> [_ N / n ]_ if ( n = 0 , A , if ( n = S , C , if ( S < n , D , B ) ) ) e. V )
89		fvmptnn04if.g	. . . 4 |- G = ( n e. NN0 |-> if ( n = 0 , A , if ( n = S , C , if ( S < n , D , B ) ) ) )
90	89	fvmpts 6285	. . 3 |- ( ( N e. NN0 /\ [_ N / n ]_ if ( n = 0 , A , if ( n = S , C , if ( S < n , D , B ) ) ) e. V ) -> ( G ` N ) = [_ N / n ]_ if ( n = 0 , A , if ( n = S , C , if ( S < n , D , B ) ) ) )
91	1, 88, 90	syl2anc 693	. 2 |- ( ph -> ( G ` N ) = [_ N / n ]_ if ( n = 0 , A , if ( n = S , C , if ( S < n , D , B ) ) ) )
92	21	eqcomd 2628	. . 3 |- ( ( ph /\ N = 0 ) -> [_ N / n ]_ A = Y )
93	34, 82	ifeqda 4121	. . . 4 |- ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) -> if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) = Y )
94	27, 93	ifeqda 4121	. . 3 |- ( ( ph /\ -. N = 0 ) -> if ( N = S , [_ N / n ]_ C , if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) ) = Y )
95	92, 94	ifeqda 4121	. 2 |- ( ph -> if ( N = 0 , [_ N / n ]_ A , if ( N = S , [_ N / n ]_ C , if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) ) ) = Y )
96	91, 20, 95	3eqtrd 2660	1 |- ( ph -> ( G ` N ) = Y )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   [.wsbc 3435   [_csb 3533   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888   RRcr 9935   0cc0 9936   < clt 10074   <_ cle 10075   NNcn 11020   NN0cn0 11292

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  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-nel 2898  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293

This theorem is referenced by:  fvmptnn04ifa  20655  fvmptnn04ifb  20656  fvmptnn04ifc  20657  fvmptnn04ifd  20658