Theorem nconstwlpolem 14096

Description: Lemma for nconstwlpo 14097. (Contributed by Jim Kingdon, 23-Jul-2024.)

Hypotheses

Ref	Expression
nconstwlpo.f	|- ( ph -> F : RR --> ZZ )
nconstwlpo.0	|- ( ph -> ( F ` 0 ) = 0 )
nconstwlpo.rp	|- ( ( ph /\ x e. RR+ ) -> ( F ` x ) =/= 0 )
nconstwlpo.g	|- ( ph -> G : NN --> { 0 , 1 } )
nconstwlpo.a	|- A = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( G ` i ) )

Assertion

Ref	Expression
nconstwlpolem	|- ( ph -> ( A. y e. NN ( G ` y ) = 0 \/ -. A. y e. NN ( G ` y ) = 0 ) )

Distinct variable groups:   x, A   y, A   x, F   y, F   i, G, y   ph, x   ph, y, i

Allowed substitution hints:   A( i)   F( i)   G( x)

Proof of Theorem nconstwlpolem

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 3993	. . . . . . . . . . . 12 |- ( x = A -> ( 0 < x <-> 0 < A ) )
2		fveq2 5496	. . . . . . . . . . . . 13 |- ( x = A -> ( F ` x ) = ( F ` A ) )
3	2	neeq1d 2358	. . . . . . . . . . . 12 |- ( x = A -> ( ( F ` x ) =/= 0 <-> ( F ` A ) =/= 0 ) )
4	1, 3	imbi12d 233	. . . . . . . . . . 11 |- ( x = A -> ( ( 0 < x -> ( F ` x ) =/= 0 ) <-> ( 0 < A -> ( F ` A ) =/= 0 ) ) )
5		elrp 9612	. . . . . . . . . . . . . 14 |- ( x e. RR+ <-> ( x e. RR /\ 0 < x ) )
6		nconstwlpo.rp	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. RR+ ) -> ( F ` x ) =/= 0 )
7	5, 6	sylan2br 286	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. RR /\ 0 < x ) ) -> ( F ` x ) =/= 0 )
8	7	expr 373	. . . . . . . . . . . 12 |- ( ( ph /\ x e. RR ) -> ( 0 < x -> ( F ` x ) =/= 0 ) )
9	8	ralrimiva 2543	. . . . . . . . . . 11 |- ( ph -> A. x e. RR ( 0 < x -> ( F ` x ) =/= 0 ) )
10		nconstwlpo.g	. . . . . . . . . . . 12 |- ( ph -> G : NN --> { 0 , 1 } )
11		nconstwlpo.a	. . . . . . . . . . . 12 |- A = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( G ` i ) )
12	10, 11	trilpolemcl 14069	. . . . . . . . . . 11 |- ( ph -> A e. RR )
13	4, 9, 12	rspcdva 2839	. . . . . . . . . 10 |- ( ph -> ( 0 < A -> ( F ` A ) =/= 0 ) )
14	13	necon2bd 2398	. . . . . . . . 9 |- ( ph -> ( ( F ` A ) = 0 -> -. 0 < A ) )
15	14	imp 123	. . . . . . . 8 |- ( ( ph /\ ( F ` A ) = 0 ) -> -. 0 < A )
16	10	adantr 274	. . . . . . . . . . . 12 |- ( ( ph /\ E. y e. NN ( G ` y ) = 1 ) -> G : NN --> { 0 , 1 } )
17		simpr 109	. . . . . . . . . . . . 13 |- ( ( ph /\ E. y e. NN ( G ` y ) = 1 ) -> E. y e. NN ( G ` y ) = 1 )
18		fveqeq2 5505	. . . . . . . . . . . . . 14 |- ( y = a -> ( ( G ` y ) = 1 <-> ( G ` a ) = 1 ) )
19	18	cbvrexv 2697	. . . . . . . . . . . . 13 |- ( E. y e. NN ( G ` y ) = 1 <-> E. a e. NN ( G ` a ) = 1 )
20	17, 19	sylib 121	. . . . . . . . . . . 12 |- ( ( ph /\ E. y e. NN ( G ` y ) = 1 ) -> E. a e. NN ( G ` a ) = 1 )
21	16, 11, 20	nconstwlpolemgt0 14095	. . . . . . . . . . 11 |- ( ( ph /\ E. y e. NN ( G ` y ) = 1 ) -> 0 < A )
22	21	ex 114	. . . . . . . . . 10 |- ( ph -> ( E. y e. NN ( G ` y ) = 1 -> 0 < A ) )
23	22	con3d 626	. . . . . . . . 9 |- ( ph -> ( -. 0 < A -> -. E. y e. NN ( G ` y ) = 1 ) )
24	23	adantr 274	. . . . . . . 8 |- ( ( ph /\ ( F ` A ) = 0 ) -> ( -. 0 < A -> -. E. y e. NN ( G ` y ) = 1 ) )
25	15, 24	mpd 13	. . . . . . 7 |- ( ( ph /\ ( F ` A ) = 0 ) -> -. E. y e. NN ( G ` y ) = 1 )
26		ralnex 2458	. . . . . . 7 |- ( A. y e. NN -. ( G ` y ) = 1 <-> -. E. y e. NN ( G ` y ) = 1 )
27	25, 26	sylibr 133	. . . . . 6 |- ( ( ph /\ ( F ` A ) = 0 ) -> A. y e. NN -. ( G ` y ) = 1 )
28	27	r19.21bi 2558	. . . . 5 |- ( ( ( ph /\ ( F ` A ) = 0 ) /\ y e. NN ) -> -. ( G ` y ) = 1 )
29	10	ad2antrr 485	. . . . . . 7 |- ( ( ( ph /\ ( F ` A ) = 0 ) /\ y e. NN ) -> G : NN --> { 0 , 1 } )
30		simpr 109	. . . . . . 7 |- ( ( ( ph /\ ( F ` A ) = 0 ) /\ y e. NN ) -> y e. NN )
31	29, 30	ffvelrnd 5632	. . . . . 6 |- ( ( ( ph /\ ( F ` A ) = 0 ) /\ y e. NN ) -> ( G ` y ) e. { 0 , 1 } )
32		elpri 3606	. . . . . 6 |- ( ( G ` y ) e. { 0 , 1 } -> ( ( G ` y ) = 0 \/ ( G ` y ) = 1 ) )
33	31, 32	syl 14	. . . . 5 |- ( ( ( ph /\ ( F ` A ) = 0 ) /\ y e. NN ) -> ( ( G ` y ) = 0 \/ ( G ` y ) = 1 ) )
34	28, 33	ecased 1344	. . . 4 |- ( ( ( ph /\ ( F ` A ) = 0 ) /\ y e. NN ) -> ( G ` y ) = 0 )
35	34	ralrimiva 2543	. . 3 |- ( ( ph /\ ( F ` A ) = 0 ) -> A. y e. NN ( G ` y ) = 0 )
36	35	orcd 728	. 2 |- ( ( ph /\ ( F ` A ) = 0 ) -> ( A. y e. NN ( G ` y ) = 0 \/ -. A. y e. NN ( G ` y ) = 0 ) )
37	10	adantr 274	. . . . . . . . 9 |- ( ( ph /\ A. y e. NN ( G ` y ) = 0 ) -> G : NN --> { 0 , 1 } )
38		simpr 109	. . . . . . . . 9 |- ( ( ph /\ A. y e. NN ( G ` y ) = 0 ) -> A. y e. NN ( G ` y ) = 0 )
39	37, 11, 38	nconstwlpolem0 14094	. . . . . . . 8 |- ( ( ph /\ A. y e. NN ( G ` y ) = 0 ) -> A = 0 )
40	39	fveq2d 5500	. . . . . . 7 |- ( ( ph /\ A. y e. NN ( G ` y ) = 0 ) -> ( F ` A ) = ( F ` 0 ) )
41		nconstwlpo.0	. . . . . . . 8 |- ( ph -> ( F ` 0 ) = 0 )
42	41	adantr 274	. . . . . . 7 |- ( ( ph /\ A. y e. NN ( G ` y ) = 0 ) -> ( F ` 0 ) = 0 )
43	40, 42	eqtrd 2203	. . . . . 6 |- ( ( ph /\ A. y e. NN ( G ` y ) = 0 ) -> ( F ` A ) = 0 )
44	43	ex 114	. . . . 5 |- ( ph -> ( A. y e. NN ( G ` y ) = 0 -> ( F ` A ) = 0 ) )
45	44	con3d 626	. . . 4 |- ( ph -> ( -. ( F ` A ) = 0 -> -. A. y e. NN ( G ` y ) = 0 ) )
46	45	imp 123	. . 3 |- ( ( ph /\ -. ( F ` A ) = 0 ) -> -. A. y e. NN ( G ` y ) = 0 )
47	46	olcd 729	. 2 |- ( ( ph /\ -. ( F ` A ) = 0 ) -> ( A. y e. NN ( G ` y ) = 0 \/ -. A. y e. NN ( G ` y ) = 0 ) )
48		nconstwlpo.f	. . . . 5 |- ( ph -> F : RR --> ZZ )
49	48, 12	ffvelrnd 5632	. . . 4 |- ( ph -> ( F ` A ) e. ZZ )
50		0z 9223	. . . 4 |- 0 e. ZZ
51		zdceq 9287	. . . 4 |- ( ( ( F ` A ) e. ZZ /\ 0 e. ZZ ) -> DECID ( F ` A ) = 0 )
52	49, 50, 51	sylancl 411	. . 3 |- ( ph -> DECID ( F ` A ) = 0 )
53		exmiddc 831	. . 3 |- (DECID ( F ` A ) = 0 -> ( ( F ` A ) = 0 \/ -. ( F ` A ) = 0 ) )
54	52, 53	syl 14	. 2 |- ( ph -> ( ( F ` A ) = 0 \/ -. ( F ` A ) = 0 ) )
55	36, 47, 54	mpjaodan 793	1 |- ( ph -> ( A. y e. NN ( G ` y ) = 0 \/ -. A. y e. NN ( G ` y ) = 0 ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 703  DECID wdc 829   = wceq 1348   e. wcel 2141   =/= wne 2340   A.wral 2448   E.wrex 2449   {cpr 3584   class class class wbr 3989   -->wf 5194   ` cfv 5198  (class class class)co 5853   RRcr 7773   0cc0 7774   1c1 7775   x. cmul 7779   < clt 7954   / cdiv 8589   NNcn 8878   2c2 8929   ZZcz 9212   RR+crp 9610   ^cexp 10475   sum_csu 11316

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-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  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-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  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-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-ico 9851  df-fz 9966  df-fzo 10099  df-seqfrec 10402  df-exp 10476  df-ihash 10710  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-sumdc 11317

This theorem is referenced by:  nconstwlpo  14097