Theorem nconstwlpolem 15822

Description: Lemma for nconstwlpo 15823. (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 4038	. . . . . . . . . . . 12 |- ( x = A -> ( 0 < x <-> 0 < A ) )
2		fveq2 5561	. . . . . . . . . . . . 13 |- ( x = A -> ( F ` x ) = ( F ` A ) )
3	2	neeq1d 2385	. . . . . . . . . . . 12 |- ( x = A -> ( ( F ` x ) =/= 0 <-> ( F ` A ) =/= 0 ) )
4	1, 3	imbi12d 234	. . . . . . . . . . 11 |- ( x = A -> ( ( 0 < x -> ( F ` x ) =/= 0 ) <-> ( 0 < A -> ( F ` A ) =/= 0 ) ) )
5		elrp 9749	. . . . . . . . . . . . . 14 |- ( x e. RR+ <-> ( x e. RR /\ 0 < x ) )
6		nconstwlpo.rp	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. RR+ ) -> ( F ` x ) =/= 0 )
7	5, 6	sylan2br 288	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. RR /\ 0 < x ) ) -> ( F ` x ) =/= 0 )
8	7	expr 375	. . . . . . . . . . . 12 |- ( ( ph /\ x e. RR ) -> ( 0 < x -> ( F ` x ) =/= 0 ) )
9	8	ralrimiva 2570	. . . . . . . . . . 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 15794	. . . . . . . . . . 11 |- ( ph -> A e. RR )
13	4, 9, 12	rspcdva 2873	. . . . . . . . . 10 |- ( ph -> ( 0 < A -> ( F ` A ) =/= 0 ) )
14	13	necon2bd 2425	. . . . . . . . 9 |- ( ph -> ( ( F ` A ) = 0 -> -. 0 < A ) )
15	14	imp 124	. . . . . . . 8 |- ( ( ph /\ ( F ` A ) = 0 ) -> -. 0 < A )
16	10	adantr 276	. . . . . . . . . . . 12 |- ( ( ph /\ E. y e. NN ( G ` y ) = 1 ) -> G : NN --> { 0 , 1 } )
17		simpr 110	. . . . . . . . . . . . 13 |- ( ( ph /\ E. y e. NN ( G ` y ) = 1 ) -> E. y e. NN ( G ` y ) = 1 )
18		fveqeq2 5570	. . . . . . . . . . . . . 14 |- ( y = a -> ( ( G ` y ) = 1 <-> ( G ` a ) = 1 ) )
19	18	cbvrexv 2730	. . . . . . . . . . . . 13 |- ( E. y e. NN ( G ` y ) = 1 <-> E. a e. NN ( G ` a ) = 1 )
20	17, 19	sylib 122	. . . . . . . . . . . 12 |- ( ( ph /\ E. y e. NN ( G ` y ) = 1 ) -> E. a e. NN ( G ` a ) = 1 )
21	16, 11, 20	nconstwlpolemgt0 15821	. . . . . . . . . . 11 |- ( ( ph /\ E. y e. NN ( G ` y ) = 1 ) -> 0 < A )
22	21	ex 115	. . . . . . . . . 10 |- ( ph -> ( E. y e. NN ( G ` y ) = 1 -> 0 < A ) )
23	22	con3d 632	. . . . . . . . 9 |- ( ph -> ( -. 0 < A -> -. E. y e. NN ( G ` y ) = 1 ) )
24	23	adantr 276	. . . . . . . 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 2485	. . . . . . 7 |- ( A. y e. NN -. ( G ` y ) = 1 <-> -. E. y e. NN ( G ` y ) = 1 )
27	25, 26	sylibr 134	. . . . . 6 |- ( ( ph /\ ( F ` A ) = 0 ) -> A. y e. NN -. ( G ` y ) = 1 )
28	27	r19.21bi 2585	. . . . 5 |- ( ( ( ph /\ ( F ` A ) = 0 ) /\ y e. NN ) -> -. ( G ` y ) = 1 )
29	10	ad2antrr 488	. . . . . . 7 |- ( ( ( ph /\ ( F ` A ) = 0 ) /\ y e. NN ) -> G : NN --> { 0 , 1 } )
30		simpr 110	. . . . . . 7 |- ( ( ( ph /\ ( F ` A ) = 0 ) /\ y e. NN ) -> y e. NN )
31	29, 30	ffvelcdmd 5701	. . . . . 6 |- ( ( ( ph /\ ( F ` A ) = 0 ) /\ y e. NN ) -> ( G ` y ) e. { 0 , 1 } )
32		elpri 3646	. . . . . 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 1360	. . . 4 |- ( ( ( ph /\ ( F ` A ) = 0 ) /\ y e. NN ) -> ( G ` y ) = 0 )
35	34	ralrimiva 2570	. . 3 |- ( ( ph /\ ( F ` A ) = 0 ) -> A. y e. NN ( G ` y ) = 0 )
36	35	orcd 734	. 2 |- ( ( ph /\ ( F ` A ) = 0 ) -> ( A. y e. NN ( G ` y ) = 0 \/ -. A. y e. NN ( G ` y ) = 0 ) )
37	10	adantr 276	. . . . . . . . 9 |- ( ( ph /\ A. y e. NN ( G ` y ) = 0 ) -> G : NN --> { 0 , 1 } )
38		simpr 110	. . . . . . . . 9 |- ( ( ph /\ A. y e. NN ( G ` y ) = 0 ) -> A. y e. NN ( G ` y ) = 0 )
39	37, 11, 38	nconstwlpolem0 15820	. . . . . . . 8 |- ( ( ph /\ A. y e. NN ( G ` y ) = 0 ) -> A = 0 )
40	39	fveq2d 5565	. . . . . . 7 |- ( ( ph /\ A. y e. NN ( G ` y ) = 0 ) -> ( F ` A ) = ( F ` 0 ) )
41		nconstwlpo.0	. . . . . . . 8 |- ( ph -> ( F ` 0 ) = 0 )
42	41	adantr 276	. . . . . . 7 |- ( ( ph /\ A. y e. NN ( G ` y ) = 0 ) -> ( F ` 0 ) = 0 )
43	40, 42	eqtrd 2229	. . . . . 6 |- ( ( ph /\ A. y e. NN ( G ` y ) = 0 ) -> ( F ` A ) = 0 )
44	43	ex 115	. . . . 5 |- ( ph -> ( A. y e. NN ( G ` y ) = 0 -> ( F ` A ) = 0 ) )
45	44	con3d 632	. . . 4 |- ( ph -> ( -. ( F ` A ) = 0 -> -. A. y e. NN ( G ` y ) = 0 ) )
46	45	imp 124	. . 3 |- ( ( ph /\ -. ( F ` A ) = 0 ) -> -. A. y e. NN ( G ` y ) = 0 )
47	46	olcd 735	. 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	ffvelcdmd 5701	. . . 4 |- ( ph -> ( F ` A ) e. ZZ )
50		0z 9356	. . . 4 |- 0 e. ZZ
51		zdceq 9420	. . . 4 |- ( ( ( F ` A ) e. ZZ /\ 0 e. ZZ ) -> DECID ( F ` A ) = 0 )
52	49, 50, 51	sylancl 413	. . 3 |- ( ph -> DECID ( F ` A ) = 0 )
53		exmiddc 837	. . 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 799	1 |- ( ph -> ( A. y e. NN ( G ` y ) = 0 \/ -. A. y e. NN ( G ` y ) = 0 ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 709  DECID wdc 835   = wceq 1364   e. wcel 2167   =/= wne 2367   A.wral 2475   E.wrex 2476   {cpr 3624   class class class wbr 4034   -->wf 5255   ` cfv 5259  (class class class)co 5925   RRcr 7897   0cc0 7898   1c1 7899   x. cmul 7903   < clt 8080   / cdiv 8718   NNcn 9009   2c2 9060   ZZcz 9345   RR+crp 9747   ^cexp 10649   sum_csu 11537

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7989  ax-resscn 7990  ax-1cn 7991  ax-1re 7992  ax-icn 7993  ax-addcl 7994  ax-addrcl 7995  ax-mulcl 7996  ax-mulrcl 7997  ax-addcom 7998  ax-mulcom 7999  ax-addass 8000  ax-mulass 8001  ax-distr 8002  ax-i2m1 8003  ax-0lt1 8004  ax-1rid 8005  ax-0id 8006  ax-rnegex 8007  ax-precex 8008  ax-cnre 8009  ax-pre-ltirr 8010  ax-pre-ltwlin 8011  ax-pre-lttrn 8012  ax-pre-apti 8013  ax-pre-ltadd 8014  ax-pre-mulgt0 8015  ax-pre-mulext 8016  ax-arch 8017  ax-caucvg 8018

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-frec 6458  df-1o 6483  df-oadd 6487  df-er 6601  df-en 6809  df-dom 6810  df-fin 6811  df-pnf 8082  df-mnf 8083  df-xr 8084  df-ltxr 8085  df-le 8086  df-sub 8218  df-neg 8219  df-reap 8621  df-ap 8628  df-div 8719  df-inn 9010  df-2 9068  df-3 9069  df-4 9070  df-n0 9269  df-z 9346  df-uz 9621  df-q 9713  df-rp 9748  df-ico 9988  df-fz 10103  df-fzo 10237  df-seqfrec 10559  df-exp 10650  df-ihash 10887  df-cj 11026  df-re 11027  df-im 11028  df-rsqrt 11182  df-abs 11183  df-clim 11463  df-sumdc 11538

This theorem is referenced by:  nconstwlpo  15823