Theorem fvmptnn04if 19921

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 3942	. . . . 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 3318	. . . . . . 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 3942	. . . . . . 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 3318	. . . . . . . . 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 3942	. . . . . . . . 9 |- [_ N / n ]_ if ( S < n , D , B ) = if ( [. N / n ]. S < n , [_ N / n ]_ D , [_ N / n ]_ B )
9		sbcbr2g 4471	. . . . . . . . . . . 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 3803	. . . . . . . . . . . . 13 |- ( N e. NN0 -> [_ N / n ]_ n = N )
12	1, 11	syl 17	. . . . . . . . . . . 12 |- ( ph -> [_ N / n ]_ n = N )
13	12	breq2d 4427	. . . . . . . . . . 11 |- ( ph -> ( S < [_ N / n ]_ n <-> S < N ) )
14	10, 13	bitrd 261	. . . . . . . . . 10 |- ( ph -> ( [. N / n ]. S < n <-> S < N ) )
15	14	ifbid 3914	. . . . . . . . 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 2507	. . . . . . . 8 |- ( ph -> [_ N / n ]_ if ( S < n , D , B ) = if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) )
17	7, 16	ifbieq2d 3917	. . . . . . 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 2507	. . . . . 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 3917	. . . . 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 2507	. . . 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 471	. . . . . 6 |- ( ( ph /\ N = 0 ) -> Y e. V )
24	21, 23	eqeltrrd 2540	. . . . 5 |- ( ( ph /\ N = 0 ) -> [_ N / n ]_ A e. V )
25		fvmptnn04if.c	. . . . . . . . 9 |- ( ( ph /\ N = S ) -> Y = [_ N / n ]_ C )
26	25	eqcomd 2467	. . . . . . . 8 |- ( ( ph /\ N = S ) -> [_ N / n ]_ C = Y )
27	26	adantlr 726	. . . . . . 7 |- ( ( ( ph /\ -. N = 0 ) /\ N = S ) -> [_ N / n ]_ C = Y )
28	22	ad2antrr 737	. . . . . . 7 |- ( ( ( ph /\ -. N = 0 ) /\ N = S ) -> Y e. V )
29	27, 28	eqeltrd 2539	. . . . . 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 2467	. . . . . . . . . . 11 |- ( ( ph /\ S < N ) -> [_ N / n ]_ D = Y )
32	31	ex 440	. . . . . . . . . 10 |- ( ph -> ( S < N -> [_ N / n ]_ D = Y ) )
33	32	ad2antrr 737	. . . . . . . . 9 |- ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) -> ( S < N -> [_ N / n ]_ D = Y ) )
34	33	imp 435	. . . . . . . 8 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ S < N ) -> [_ N / n ]_ D = Y )
35	22	ad3antrrr 741	. . . . . . . 8 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ S < N ) -> Y e. V )
36	34, 35	eqeltrd 2539	. . . . . . 7 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ S < N ) -> [_ N / n ]_ D e. V )
37		simplll 773	. . . . . . . . 9 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) -> ph )
38		ancom 456	. . . . . . . . . . . . 13 |- ( ( -. S < N /\ ph ) <-> ( ph /\ -. S < N ) )
39	38	anbi2i 705	. . . . . . . . . . . 12 |- ( ( ( -. N = 0 /\ -. N = S ) /\ ( -. S < N /\ ph ) ) <-> ( ( -. N = 0 /\ -. N = S ) /\ ( ph /\ -. S < N ) ) )
40		ancom 456	. . . . . . . . . . . . 13 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) <-> ( ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) /\ ph ) )
41		anass 659	. . . . . . . . . . . . . . 15 |- ( ( ( -. N = 0 /\ -. N = S ) /\ -. S < N ) <-> ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) )
42	41	bicomi 207	. . . . . . . . . . . . . 14 |- ( ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) <-> ( ( -. N = 0 /\ -. N = S ) /\ -. S < N ) )
43	42	anbi1i 706	. . . . . . . . . . . . 13 |- ( ( ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) /\ ph ) <-> ( ( ( -. N = 0 /\ -. N = S ) /\ -. S < N ) /\ ph ) )
44		anass 659	. . . . . . . . . . . . 13 |- ( ( ( ( -. N = 0 /\ -. N = S ) /\ -. S < N ) /\ ph ) <-> ( ( -. N = 0 /\ -. N = S ) /\ ( -. S < N /\ ph ) ) )
45	40, 43, 44	3bitri 279	. . . . . . . . . . . 12 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) <-> ( ( -. N = 0 /\ -. N = S ) /\ ( -. S < N /\ ph ) ) )
46		anass 659	. . . . . . . . . . . 12 |- ( ( ( ( -. N = 0 /\ -. N = S ) /\ ph ) /\ -. S < N ) <-> ( ( -. N = 0 /\ -. N = S ) /\ ( ph /\ -. S < N ) ) )
47	39, 45, 46	3bitr4i 285	. . . . . . . . . . 11 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) <-> ( ( ( -. N = 0 /\ -. N = S ) /\ ph ) /\ -. S < N ) )
48		an32 812	. . . . . . . . . . . . 13 |- ( ( ( -. N = 0 /\ -. N = S ) /\ ph ) <-> ( ( -. N = 0 /\ ph ) /\ -. N = S ) )
49		ancom 456	. . . . . . . . . . . . . 14 |- ( ( -. N = 0 /\ ph ) <-> ( ph /\ -. N = 0 ) )
50	49	anbi1i 706	. . . . . . . . . . . . 13 |- ( ( ( -. N = 0 /\ ph ) /\ -. N = S ) <-> ( ( ph /\ -. N = 0 ) /\ -. N = S ) )
51	48, 50	bitri 257	. . . . . . . . . . . 12 |- ( ( ( -. N = 0 /\ -. N = S ) /\ ph ) <-> ( ( ph /\ -. N = 0 ) /\ -. N = S ) )
52	51	anbi1i 706	. . . . . . . . . . 11 |- ( ( ( ( -. N = 0 /\ -. N = S ) /\ ph ) /\ -. S < N ) <-> ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) )
53	47, 52	bitri 257	. . . . . . . . . 10 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) <-> ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) )
54		df-ne 2634	. . . . . . . . . . . . 13 |- ( N =/= 0 <-> -. N = 0 )
55		elnnne0 10911	. . . . . . . . . . . . . . 15 |- ( N e. NN <-> ( N e. NN0 /\ N =/= 0 ) )
56		nngt0 10665	. . . . . . . . . . . . . . 15 |- ( N e. NN -> 0 < N )
57	55, 56	sylbir 218	. . . . . . . . . . . . . 14 |- ( ( N e. NN0 /\ N =/= 0 ) -> 0 < N )
58	57	expcom 441	. . . . . . . . . . . . 13 |- ( N =/= 0 -> ( N e. NN0 -> 0 < N ) )
59	54, 58	sylbir 218	. . . . . . . . . . . 12 |- ( -. N = 0 -> ( N e. NN0 -> 0 < N ) )
60	59	adantr 471	. . . . . . . . . . 11 |- ( ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) -> ( N e. NN0 -> 0 < N ) )
61	1, 60	mpan9 476	. . . . . . . . . 10 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> 0 < N )
62	53, 61	sylbir 218	. . . . . . . . 9 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) -> 0 < N )
63	1	nn0red 10954	. . . . . . . . . . . . . . . 16 |- ( ph -> N e. RR )
64		fvmptnn04if.s	. . . . . . . . . . . . . . . . 17 |- ( ph -> S e. NN )
65	64	nnred 10651	. . . . . . . . . . . . . . . 16 |- ( ph -> S e. RR )
66	63, 65	lenltd 9806	. . . . . . . . . . . . . . 15 |- ( ph -> ( N <_ S <-> -. S < N ) )
67	66	biimprd 231	. . . . . . . . . . . . . 14 |- ( ph -> ( -. S < N -> N <_ S ) )
68	67	adantld 473	. . . . . . . . . . . . 13 |- ( ph -> ( ( -. N = S /\ -. S < N ) -> N <_ S ) )
69	68	adantld 473	. . . . . . . . . . . 12 |- ( ph -> ( ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) -> N <_ S ) )
70	69	imp 435	. . . . . . . . . . 11 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> N <_ S )
71		nesym 2691	. . . . . . . . . . . . . 14 |- ( S =/= N <-> -. N = S )
72	71	biimpri 211	. . . . . . . . . . . . 13 |- ( -. N = S -> S =/= N )
73	72	adantr 471	. . . . . . . . . . . 12 |- ( ( -. N = S /\ -. S < N ) -> S =/= N )
74	73	ad2antll 740	. . . . . . . . . . 11 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> S =/= N )
75	63	adantr 471	. . . . . . . . . . . 12 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> N e. RR )
76	65	adantr 471	. . . . . . . . . . . 12 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> S e. RR )
77	75, 76	ltlend 9805	. . . . . . . . . . 11 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> ( N < S <-> ( N <_ S /\ S =/= N ) ) )
78	70, 74, 77	mpbir2and 938	. . . . . . . . . 10 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> N < S )
79	53, 78	sylbir 218	. . . . . . . . 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 2467	. . . . . . . . 9 |- ( ( ph /\ 0 < N /\ N < S ) -> [_ N / n ]_ B = Y )
82	37, 62, 79, 81	syl3anc 1276	. . . . . . . 8 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) -> [_ N / n ]_ B = Y )
83	22	ad3antrrr 741	. . . . . . . 8 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) -> Y e. V )
84	82, 83	eqeltrd 2539	. . . . . . 7 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) -> [_ N / n ]_ B e. V )
85	36, 84	ifclda 3924	. . . . . 6 |- ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) -> if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) e. V )
86	29, 85	ifclda 3924	. . . . 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 3924	. . . 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 2539	. . 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 5973	. . 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 671	. 2 |- ( ph -> ( G ` N ) = [_ N / n ]_ if ( n = 0 , A , if ( n = S , C , if ( S < n , D , B ) ) ) )
92	21	eqcomd 2467	. . 3 |- ( ( ph /\ N = 0 ) -> [_ N / n ]_ A = Y )
93	34, 82	ifeqda 3925	. . . 4 |- ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) -> if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) = Y )
94	27, 93	ifeqda 3925	. . 3 |- ( ( ph /\ -. N = 0 ) -> if ( N = S , [_ N / n ]_ C , if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) ) = Y )
95	92, 94	ifeqda 3925	. 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 2499	1 |- ( ph -> ( G ` N ) = Y )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 375   /\ w3a 991   = wceq 1454   e. wcel 1897   =/= wne 2632   [.wsbc 3278   [_csb 3374   ifcif 3892   class class class wbr 4415   |-> cmpt 4474   ` cfv 5600   RRcr 9563   0cc0 9564   < clt 9700   <_ cle 9701   NNcn 10636   NN0cn0 10897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-fal 1460  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-er 7388  df-en 7595  df-dom 7596  df-sdom 7597  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-nn 10637  df-n0 10898

This theorem is referenced by:  fvmptnn04ifa  19922  fvmptnn04ifb  19923  fvmptnn04ifc  19924  fvmptnn04ifd  19925