Theorem fvmptnn04if 19223

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 3976	. . . . 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 3353	. . . . . . 7 |- ( N e. NN0 -> ( [. N / n ]. n = 0 <-> N = 0 ) )
4	1, 3	syl 16	. . . . . 6 |- ( ph -> ( [. N / n ]. n = 0 <-> N = 0 ) )
5		csbif 3976	. . . . . . 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 3353	. . . . . . . . 9 |- ( N e. NN0 -> ( [. N / n ]. n = S <-> N = S ) )
7	1, 6	syl 16	. . . . . . . 8 |- ( ph -> ( [. N / n ]. n = S <-> N = S ) )
8		csbif 3976	. . . . . . . . 9 |- [_ N / n ]_ if ( S < n , D , B ) = if ( [. N / n ]. S < n , [_ N / n ]_ D , [_ N / n ]_ B )
9		sbcbr2g 4493	. . . . . . . . . . . 12 |- ( N e. NN0 -> ( [. N / n ]. S < n <-> S < [_ N / n ]_ n ) )
10	1, 9	syl 16	. . . . . . . . . . 11 |- ( ph -> ( [. N / n ]. S < n <-> S < [_ N / n ]_ n ) )
11		csbvarg 3834	. . . . . . . . . . . . 13 |- ( N e. NN0 -> [_ N / n ]_ n = N )
12	1, 11	syl 16	. . . . . . . . . . . 12 |- ( ph -> [_ N / n ]_ n = N )
13	12	breq2d 4449	. . . . . . . . . . 11 |- ( ph -> ( S < [_ N / n ]_ n <-> S < N ) )
14	10, 13	bitrd 253	. . . . . . . . . 10 |- ( ph -> ( [. N / n ]. S < n <-> S < N ) )
15	14	ifbid 3948	. . . . . . . . 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 2496	. . . . . . . 8 |- ( ph -> [_ N / n ]_ if ( S < n , D , B ) = if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) )
17	7, 16	ifbieq2d 3951	. . . . . . 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 2496	. . . . . 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 3951	. . . . 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 2496	. . . 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 465	. . . . . 6 |- ( ( ph /\ N = 0 ) -> Y e. V )
24	21, 23	eqeltrrd 2532	. . . . 5 |- ( ( ph /\ N = 0 ) -> [_ N / n ]_ A e. V )
25		fvmptnn04if.c	. . . . . . . . 9 |- ( ( ph /\ N = S ) -> Y = [_ N / n ]_ C )
26	25	eqcomd 2451	. . . . . . . 8 |- ( ( ph /\ N = S ) -> [_ N / n ]_ C = Y )
27	26	adantlr 714	. . . . . . 7 |- ( ( ( ph /\ -. N = 0 ) /\ N = S ) -> [_ N / n ]_ C = Y )
28	22	ad2antrr 725	. . . . . . 7 |- ( ( ( ph /\ -. N = 0 ) /\ N = S ) -> Y e. V )
29	27, 28	eqeltrd 2531	. . . . . 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 2451	. . . . . . . . . . 11 |- ( ( ph /\ S < N ) -> [_ N / n ]_ D = Y )
32	31	ex 434	. . . . . . . . . 10 |- ( ph -> ( S < N -> [_ N / n ]_ D = Y ) )
33	32	ad2antrr 725	. . . . . . . . 9 |- ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) -> ( S < N -> [_ N / n ]_ D = Y ) )
34	33	imp 429	. . . . . . . 8 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ S < N ) -> [_ N / n ]_ D = Y )
35	22	ad3antrrr 729	. . . . . . . 8 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ S < N ) -> Y e. V )
36	34, 35	eqeltrd 2531	. . . . . . 7 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ S < N ) -> [_ N / n ]_ D e. V )
37		simplll 759	. . . . . . . . 9 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) -> ph )
38		ancom 450	. . . . . . . . . . . . 13 |- ( ( -. S < N /\ ph ) <-> ( ph /\ -. S < N ) )
39	38	anbi2i 694	. . . . . . . . . . . 12 |- ( ( ( -. N = 0 /\ -. N = S ) /\ ( -. S < N /\ ph ) ) <-> ( ( -. N = 0 /\ -. N = S ) /\ ( ph /\ -. S < N ) ) )
40		ancom 450	. . . . . . . . . . . . 13 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) <-> ( ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) /\ ph ) )
41		anass 649	. . . . . . . . . . . . . . 15 |- ( ( ( -. N = 0 /\ -. N = S ) /\ -. S < N ) <-> ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) )
42	41	bicomi 202	. . . . . . . . . . . . . 14 |- ( ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) <-> ( ( -. N = 0 /\ -. N = S ) /\ -. S < N ) )
43	42	anbi1i 695	. . . . . . . . . . . . 13 |- ( ( ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) /\ ph ) <-> ( ( ( -. N = 0 /\ -. N = S ) /\ -. S < N ) /\ ph ) )
44		anass 649	. . . . . . . . . . . . 13 |- ( ( ( ( -. N = 0 /\ -. N = S ) /\ -. S < N ) /\ ph ) <-> ( ( -. N = 0 /\ -. N = S ) /\ ( -. S < N /\ ph ) ) )
45	40, 43, 44	3bitri 271	. . . . . . . . . . . 12 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) <-> ( ( -. N = 0 /\ -. N = S ) /\ ( -. S < N /\ ph ) ) )
46		anass 649	. . . . . . . . . . . 12 |- ( ( ( ( -. N = 0 /\ -. N = S ) /\ ph ) /\ -. S < N ) <-> ( ( -. N = 0 /\ -. N = S ) /\ ( ph /\ -. S < N ) ) )
47	39, 45, 46	3bitr4i 277	. . . . . . . . . . 11 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) <-> ( ( ( -. N = 0 /\ -. N = S ) /\ ph ) /\ -. S < N ) )
48		an32 798	. . . . . . . . . . . . 13 |- ( ( ( -. N = 0 /\ -. N = S ) /\ ph ) <-> ( ( -. N = 0 /\ ph ) /\ -. N = S ) )
49		ancom 450	. . . . . . . . . . . . . 14 |- ( ( -. N = 0 /\ ph ) <-> ( ph /\ -. N = 0 ) )
50	49	anbi1i 695	. . . . . . . . . . . . 13 |- ( ( ( -. N = 0 /\ ph ) /\ -. N = S ) <-> ( ( ph /\ -. N = 0 ) /\ -. N = S ) )
51	48, 50	bitri 249	. . . . . . . . . . . 12 |- ( ( ( -. N = 0 /\ -. N = S ) /\ ph ) <-> ( ( ph /\ -. N = 0 ) /\ -. N = S ) )
52	51	anbi1i 695	. . . . . . . . . . 11 |- ( ( ( ( -. N = 0 /\ -. N = S ) /\ ph ) /\ -. S < N ) <-> ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) )
53	47, 52	bitri 249	. . . . . . . . . 10 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) <-> ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) )
54		df-ne 2640	. . . . . . . . . . . . 13 |- ( N =/= 0 <-> -. N = 0 )
55		elnnne0 10815	. . . . . . . . . . . . . . 15 |- ( N e. NN <-> ( N e. NN0 /\ N =/= 0 ) )
56		nngt0 10571	. . . . . . . . . . . . . . 15 |- ( N e. NN -> 0 < N )
57	55, 56	sylbir 213	. . . . . . . . . . . . . 14 |- ( ( N e. NN0 /\ N =/= 0 ) -> 0 < N )
58	57	expcom 435	. . . . . . . . . . . . 13 |- ( N =/= 0 -> ( N e. NN0 -> 0 < N ) )
59	54, 58	sylbir 213	. . . . . . . . . . . 12 |- ( -. N = 0 -> ( N e. NN0 -> 0 < N ) )
60	59	adantr 465	. . . . . . . . . . 11 |- ( ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) -> ( N e. NN0 -> 0 < N ) )
61	1, 60	mpan9 469	. . . . . . . . . 10 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> 0 < N )
62	53, 61	sylbir 213	. . . . . . . . 9 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) -> 0 < N )
63	1	nn0red 10859	. . . . . . . . . . . . . . . 16 |- ( ph -> N e. RR )
64		fvmptnn04if.s	. . . . . . . . . . . . . . . . 17 |- ( ph -> S e. NN )
65	64	nnred 10557	. . . . . . . . . . . . . . . 16 |- ( ph -> S e. RR )
66	63, 65	lenltd 9734	. . . . . . . . . . . . . . 15 |- ( ph -> ( N <_ S <-> -. S < N ) )
67	66	biimprd 223	. . . . . . . . . . . . . 14 |- ( ph -> ( -. S < N -> N <_ S ) )
68	67	adantld 467	. . . . . . . . . . . . 13 |- ( ph -> ( ( -. N = S /\ -. S < N ) -> N <_ S ) )
69	68	adantld 467	. . . . . . . . . . . 12 |- ( ph -> ( ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) -> N <_ S ) )
70	69	imp 429	. . . . . . . . . . 11 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> N <_ S )
71		nesym 2715	. . . . . . . . . . . . . 14 |- ( S =/= N <-> -. N = S )
72	71	biimpri 206	. . . . . . . . . . . . 13 |- ( -. N = S -> S =/= N )
73	72	adantr 465	. . . . . . . . . . . 12 |- ( ( -. N = S /\ -. S < N ) -> S =/= N )
74	73	ad2antll 728	. . . . . . . . . . 11 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> S =/= N )
75	63	adantr 465	. . . . . . . . . . . 12 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> N e. RR )
76	65	adantr 465	. . . . . . . . . . . 12 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> S e. RR )
77	75, 76	ltlend 9733	. . . . . . . . . . 11 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> ( N < S <-> ( N <_ S /\ S =/= N ) ) )
78	70, 74, 77	mpbir2and 922	. . . . . . . . . 10 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> N < S )
79	53, 78	sylbir 213	. . . . . . . . 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 2451	. . . . . . . . 9 |- ( ( ph /\ 0 < N /\ N < S ) -> [_ N / n ]_ B = Y )
82	37, 62, 79, 81	syl3anc 1229	. . . . . . . 8 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) -> [_ N / n ]_ B = Y )
83	22	ad3antrrr 729	. . . . . . . 8 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) -> Y e. V )
84	82, 83	eqeltrd 2531	. . . . . . 7 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) -> [_ N / n ]_ B e. V )
85	36, 84	ifclda 3958	. . . . . 6 |- ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) -> if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) e. V )
86	29, 85	ifclda 3958	. . . . 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 3958	. . . 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 2531	. . 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 5943	. . 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 661	. 2 |- ( ph -> ( G ` N ) = [_ N / n ]_ if ( n = 0 , A , if ( n = S , C , if ( S < n , D , B ) ) ) )
92	21	eqcomd 2451	. . 3 |- ( ( ph /\ N = 0 ) -> [_ N / n ]_ A = Y )
93	34, 82	ifeqda 3959	. . . 4 |- ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) -> if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) = Y )
94	27, 93	ifeqda 3959	. . 3 |- ( ( ph /\ -. N = 0 ) -> if ( N = S , [_ N / n ]_ C , if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) ) = Y )
95	92, 94	ifeqda 3959	. 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 2488	1 |- ( ph -> ( G ` N ) = Y )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 974   = wceq 1383   e. wcel 1804   =/= wne 2638   [.wsbc 3313   [_csb 3420   ifcif 3926   class class class wbr 4437   |-> cmpt 4495   ` cfv 5578   RRcr 9494   0cc0 9495   < clt 9631   <_ cle 9632   NNcn 10542   NN0cn0 10801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-n0 10802

This theorem is referenced by:  fvmptnn04ifa  19224  fvmptnn04ifb  19225  fvmptnn04ifc  19226  fvmptnn04ifd  19227