Theorem nconstwlpolem 16977

Description: Lemma for nconstwlpo 16978. (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 4118	. . . . . . . . . . . 12 |- ( x = A -> ( 0 < x <-> 0 < A ) )
2		fveq2 5675	. . . . . . . . . . . . 13 |- ( x = A -> ( F ` x ) = ( F ` A ) )
3	2	neeq1d 2432	. . . . . . . . . . . 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 10006	. . . . . . . . . . . . . 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 2617	. . . . . . . . . . 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 16947	. . . . . . . . . . 11 |- ( ph -> A e. RR )
13	4, 9, 12	rspcdva 2928	. . . . . . . . . 10 |- ( ph -> ( 0 < A -> ( F ` A ) =/= 0 ) )
14	13	necon2bd 2472	. . . . . . . . 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 5684	. . . . . . . . . . . . . 14 |- ( y = a -> ( ( G ` y ) = 1 <-> ( G ` a ) = 1 ) )
19	18	cbvrexv 2781	. . . . . . . . . . . . 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 16976	. . . . . . . . . . 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 636	. . . . . . . . 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 2532	. . . . . . 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 2632	. . . . 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 5818	. . . . . 6 |- ( ( ( ph /\ ( F ` A ) = 0 ) /\ y e. NN ) -> ( G ` y ) e. { 0 , 1 } )
32		elpri 3717	. . . . . 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 1386	. . . 4 |- ( ( ( ph /\ ( F ` A ) = 0 ) /\ y e. NN ) -> ( G ` y ) = 0 )
35	34	ralrimiva 2617	. . 3 |- ( ( ph /\ ( F ` A ) = 0 ) -> A. y e. NN ( G ` y ) = 0 )
36	35	orcd 741	. 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 16975	. . . . . . . 8 |- ( ( ph /\ A. y e. NN ( G ` y ) = 0 ) -> A = 0 )
40	39	fveq2d 5679	. . . . . . 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 2267	. . . . . 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 636	. . . 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 742	. 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 5818	. . . 4 |- ( ph -> ( F ` A ) e. ZZ )
50		0z 9605	. . . 4 |- 0 e. ZZ
51		zdceq 9670	. . . 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 844	. . 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 806	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 716  DECID wdc 842   = wceq 1398   e. wcel 2205   =/= wne 2414   A.wral 2522   E.wrex 2523   {cpr 3695   class class class wbr 4114   -->wf 5353   ` cfv 5357  (class class class)co 6058   RRcr 8142   0cc0 8143   1c1 8144   x. cmul 8148   < clt 8324   / cdiv 8963   NNcn 9254   2c2 9305   ZZcz 9594   RR+crp 10004   ^cexp 10924   sum_csu 12063

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-oadd 6664  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-ico 10246  df-fz 10362  df-fzo 10499  df-seqfrec 10834  df-exp 10925  df-ihash 11164  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-clim 11989  df-sumdc 12064

This theorem is referenced by:  nconstwlpo  16978