Theorem nconstwlpolem 17020

Description: Lemma for nconstwlpo 17021. (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 4129	. . . . . . . . . . . 12 |- ( x = A -> ( 0 < x <-> 0 < A ) )
2		fveq2 5690	. . . . . . . . . . . . 13 |- ( x = A -> ( F ` x ) = ( F ` A ) )
3	2	neeq1d 2438	. . . . . . . . . . . 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 10035	. . . . . . . . . . . . . 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 2623	. . . . . . . . . . 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 16991	. . . . . . . . . . 11 |- ( ph -> A e. RR )
13	4, 9, 12	rspcdva 2934	. . . . . . . . . 10 |- ( ph -> ( 0 < A -> ( F ` A ) =/= 0 ) )
14	13	necon2bd 2478	. . . . . . . . 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 5699	. . . . . . . . . . . . . 14 |- ( y = a -> ( ( G ` y ) = 1 <-> ( G ` a ) = 1 ) )
19	18	cbvrexv 2787	. . . . . . . . . . . . 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 17019	. . . . . . . . . . 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 640	. . . . . . . . 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 2538	. . . . . . 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 2638	. . . . 5 |- ( ( ( ph /\ ( F ` A ) = 0 ) /\ y e. NN ) -> -. ( G ` y ) = 1 )
29	10	ad2antrr 492	. . . . . . 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 5835	. . . . . 6 |- ( ( ( ph /\ ( F ` A ) = 0 ) /\ y e. NN ) -> ( G ` y ) e. { 0 , 1 } )
32		elpri 3728	. . . . . 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 1390	. . . 4 |- ( ( ( ph /\ ( F ` A ) = 0 ) /\ y e. NN ) -> ( G ` y ) = 0 )
35	34	ralrimiva 2623	. . 3 |- ( ( ph /\ ( F ` A ) = 0 ) -> A. y e. NN ( G ` y ) = 0 )
36	35	orcd 745	. 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 17018	. . . . . . . 8 |- ( ( ph /\ A. y e. NN ( G ` y ) = 0 ) -> A = 0 )
40	39	fveq2d 5694	. . . . . . 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 2271	. . . . . 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 640	. . . 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 746	. 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 5835	. . . 4 |- ( ph -> ( F ` A ) e. ZZ )
50		0z 9634	. . . 4 |- 0 e. ZZ
51		zdceq 9699	. . . 4 |- ( ( ( F ` A ) e. ZZ /\ 0 e. ZZ ) -> DECID ( F ` A ) = 0 )
52	49, 50, 51	sylancl 417	. . 3 |- ( ph -> DECID ( F ` A ) = 0 )
53		exmiddc 848	. . 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 810	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 720  DECID wdc 846   = wceq 1402   e. wcel 2209   =/= wne 2420   A.wral 2528   E.wrex 2529   {cpr 3706   class class class wbr 4125   -->wf 5368   ` cfv 5372  (class class class)co 6075   RRcr 8168   0cc0 8169   1c1 8170   x. cmul 8174   < clt 8350   / cdiv 8992   NNcn 9283   2c2 9334   ZZcz 9623   RR+crp 10033   ^cexp 10953   sum_csu 12097

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-1o 6677  df-oadd 6681  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-ico 10275  df-fz 10391  df-fzo 10528  df-seqfrec 10863  df-exp 10954  df-ihash 11193  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-clim 12023  df-sumdc 12098

This theorem is referenced by:  nconstwlpo  17021