Theorem nconstwlpolem 16798

Description: Lemma for nconstwlpo 16799. (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 4097	. . . . . . . . . . . 12 |- ( x = A -> ( 0 < x <-> 0 < A ) )
2		fveq2 5648	. . . . . . . . . . . . 13 |- ( x = A -> ( F ` x ) = ( F ` A ) )
3	2	neeq1d 2421	. . . . . . . . . . . 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 9951	. . . . . . . . . . . . . 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 2606	. . . . . . . . . . 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 16769	. . . . . . . . . . 11 |- ( ph -> A e. RR )
13	4, 9, 12	rspcdva 2916	. . . . . . . . . 10 |- ( ph -> ( 0 < A -> ( F ` A ) =/= 0 ) )
14	13	necon2bd 2461	. . . . . . . . 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 5657	. . . . . . . . . . . . . 14 |- ( y = a -> ( ( G ` y ) = 1 <-> ( G ` a ) = 1 ) )
19	18	cbvrexv 2769	. . . . . . . . . . . . 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 16797	. . . . . . . . . . 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 2521	. . . . . . 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 2621	. . . . 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 5791	. . . . . 6 |- ( ( ( ph /\ ( F ` A ) = 0 ) /\ y e. NN ) -> ( G ` y ) e. { 0 , 1 } )
32		elpri 3696	. . . . . 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 2606	. . 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 16796	. . . . . . . 8 |- ( ( ph /\ A. y e. NN ( G ` y ) = 0 ) -> A = 0 )
40	39	fveq2d 5652	. . . . . . 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 2264	. . . . . 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 5791	. . . 4 |- ( ph -> ( F ` A ) e. ZZ )
50		0z 9551	. . . 4 |- 0 e. ZZ
51		zdceq 9616	. . . 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 2202   =/= wne 2403   A.wral 2511   E.wrex 2512   {cpr 3674   class class class wbr 4093   -->wf 5329   ` cfv 5333  (class class class)co 6028   RRcr 8091   0cc0 8092   1c1 8093   x. cmul 8097   < clt 8273   / cdiv 8911   NNcn 9202   2c2 9253   ZZcz 9540   RR+crp 9949   ^cexp 10863   sum_csu 11993

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211  ax-caucvg 8212

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-frec 6600  df-1o 6625  df-oadd 6629  df-er 6745  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-n0 9462  df-z 9541  df-uz 9817  df-q 9915  df-rp 9950  df-ico 10190  df-fz 10306  df-fzo 10440  df-seqfrec 10773  df-exp 10864  df-ihash 11101  df-cj 11482  df-re 11483  df-im 11484  df-rsqrt 11638  df-abs 11639  df-clim 11919  df-sumdc 11994

This theorem is referenced by:  nconstwlpo  16799