Theorem resqrexlemfp1 11031

Description: Lemma for resqrex 11048. Recursion rule. This sequence is the ancient method for computing square roots, often known as the babylonian method, although known to many ancient cultures. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)

Hypotheses

Ref	Expression
resqrexlemex.seq	|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) )
resqrexlemex.a	|- ( ph -> A e. RR )
resqrexlemex.agt0	|- ( ph -> 0 <_ A )

Assertion

Ref	Expression
resqrexlemfp1	|- ( ( ph /\ N e. NN ) -> ( F ` ( N + 1 ) ) = ( ( ( F ` N ) + ( A / ( F ` N ) ) ) / 2 ) )

Distinct variable groups:   y, A, z   ph, y, z

Allowed substitution hints:   F( y, z)   N( y, z)

Proof of Theorem resqrexlemfp1

Dummy variables a b c d are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elnnuz 9577	. . . . . 6 |- ( N e. NN <-> N e. ( ZZ>= ` 1 ) )
2	1	biimpi 120	. . . . 5 |- ( N e. NN -> N e. ( ZZ>= ` 1 ) )
3	2	adantl 277	. . . 4 |- ( ( ph /\ N e. NN ) -> N e. ( ZZ>= ` 1 ) )
4		elnnuz 9577	. . . . . 6 |- ( a e. NN <-> a e. ( ZZ>= ` 1 ) )
5		resqrexlemex.a	. . . . . . 7 |- ( ph -> A e. RR )
6		resqrexlemex.agt0	. . . . . . 7 |- ( ph -> 0 <_ A )
7	5, 6	resqrexlem1arp 11027	. . . . . 6 |- ( ( ph /\ a e. NN ) -> ( ( NN X. { ( 1 + A ) } ) ` a ) e. RR+ )
8	4, 7	sylan2br 288	. . . . 5 |- ( ( ph /\ a e. ( ZZ>= ` 1 ) ) -> ( ( NN X. { ( 1 + A ) } ) ` a ) e. RR+ )
9	8	adantlr 477	. . . 4 |- ( ( ( ph /\ N e. NN ) /\ a e. ( ZZ>= ` 1 ) ) -> ( ( NN X. { ( 1 + A ) } ) ` a ) e. RR+ )
10	5, 6	resqrexlemp1rp 11028	. . . . 5 |- ( ( ph /\ ( a e. RR+ /\ b e. RR+ ) ) -> ( a ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) b ) e. RR+ )
11	10	adantlr 477	. . . 4 |- ( ( ( ph /\ N e. NN ) /\ ( a e. RR+ /\ b e. RR+ ) ) -> ( a ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) b ) e. RR+ )
12	3, 9, 11	seq3p1 10475	. . 3 |- ( ( ph /\ N e. NN ) -> ( seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) ) ` ( N + 1 ) ) = ( ( seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) ) ` N ) ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) ( ( NN X. { ( 1 + A ) } ) ` ( N + 1 ) ) ) )
13		resqrexlemex.seq	. . . 4 |- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) )
14	13	fveq1i 5528	. . 3 |- ( F ` ( N + 1 ) ) = ( seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) ) ` ( N + 1 ) )
15	13	fveq1i 5528	. . . 4 |- ( F ` N ) = ( seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) ) ` N )
16	15	oveq1i 5898	. . 3 |- ( ( F ` N ) ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) ( ( NN X. { ( 1 + A ) } ) ` ( N + 1 ) ) ) = ( ( seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) ) ` N ) ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) ( ( NN X. { ( 1 + A ) } ) ` ( N + 1 ) ) )
17	12, 14, 16	3eqtr4g 2245	. 2 |- ( ( ph /\ N e. NN ) -> ( F ` ( N + 1 ) ) = ( ( F ` N ) ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) ( ( NN X. { ( 1 + A ) } ) ` ( N + 1 ) ) ) )
18		id 19	. . . . . . 7 |- ( y = c -> y = c )
19		oveq2 5896	. . . . . . 7 |- ( y = c -> ( A / y ) = ( A / c ) )
20	18, 19	oveq12d 5906	. . . . . 6 |- ( y = c -> ( y + ( A / y ) ) = ( c + ( A / c ) ) )
21	20	oveq1d 5903	. . . . 5 |- ( y = c -> ( ( y + ( A / y ) ) / 2 ) = ( ( c + ( A / c ) ) / 2 ) )
22		eqidd 2188	. . . . 5 |- ( z = d -> ( ( c + ( A / c ) ) / 2 ) = ( ( c + ( A / c ) ) / 2 ) )
23	21, 22	cbvmpov 5968	. . . 4 |- ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) = ( c e. RR+ , d e. RR+ |-> ( ( c + ( A / c ) ) / 2 ) )
24	23	a1i 9	. . 3 |- ( ( ph /\ N e. NN ) -> ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) = ( c e. RR+ , d e. RR+ |-> ( ( c + ( A / c ) ) / 2 ) ) )
25		id 19	. . . . . 6 |- ( c = ( F ` N ) -> c = ( F ` N ) )
26		oveq2 5896	. . . . . 6 |- ( c = ( F ` N ) -> ( A / c ) = ( A / ( F ` N ) ) )
27	25, 26	oveq12d 5906	. . . . 5 |- ( c = ( F ` N ) -> ( c + ( A / c ) ) = ( ( F ` N ) + ( A / ( F ` N ) ) ) )
28	27	oveq1d 5903	. . . 4 |- ( c = ( F ` N ) -> ( ( c + ( A / c ) ) / 2 ) = ( ( ( F ` N ) + ( A / ( F ` N ) ) ) / 2 ) )
29	28	ad2antrl 490	. . 3 |- ( ( ( ph /\ N e. NN ) /\ ( c = ( F ` N ) /\ d = ( ( NN X. { ( 1 + A ) } ) ` ( N + 1 ) ) ) ) -> ( ( c + ( A / c ) ) / 2 ) = ( ( ( F ` N ) + ( A / ( F ` N ) ) ) / 2 ) )
30	13, 5, 6	resqrexlemf 11029	. . . 4 |- ( ph -> F : NN --> RR+ )
31	30	ffvelcdmda 5664	. . 3 |- ( ( ph /\ N e. NN ) -> ( F ` N ) e. RR+ )
32		peano2nn 8944	. . . 4 |- ( N e. NN -> ( N + 1 ) e. NN )
33	5, 6	resqrexlem1arp 11027	. . . 4 |- ( ( ph /\ ( N + 1 ) e. NN ) -> ( ( NN X. { ( 1 + A ) } ) ` ( N + 1 ) ) e. RR+ )
34	32, 33	sylan2 286	. . 3 |- ( ( ph /\ N e. NN ) -> ( ( NN X. { ( 1 + A ) } ) ` ( N + 1 ) ) e. RR+ )
35	31	rpred 9709	. . . . 5 |- ( ( ph /\ N e. NN ) -> ( F ` N ) e. RR )
36	5	adantr 276	. . . . . 6 |- ( ( ph /\ N e. NN ) -> A e. RR )
37	36, 31	rerpdivcld 9741	. . . . 5 |- ( ( ph /\ N e. NN ) -> ( A / ( F ` N ) ) e. RR )
38	35, 37	readdcld 8000	. . . 4 |- ( ( ph /\ N e. NN ) -> ( ( F ` N ) + ( A / ( F ` N ) ) ) e. RR )
39	38	rehalfcld 9178	. . 3 |- ( ( ph /\ N e. NN ) -> ( ( ( F ` N ) + ( A / ( F ` N ) ) ) / 2 ) e. RR )
40	24, 29, 31, 34, 39	ovmpod 6015	. 2 |- ( ( ph /\ N e. NN ) -> ( ( F ` N ) ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) ( ( NN X. { ( 1 + A ) } ) ` ( N + 1 ) ) ) = ( ( ( F ` N ) + ( A / ( F ` N ) ) ) / 2 ) )
41	17, 40	eqtrd 2220	1 |- ( ( ph /\ N e. NN ) -> ( F ` ( N + 1 ) ) = ( ( ( F ` N ) + ( A / ( F ` N ) ) ) / 2 ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1363   e. wcel 2158   {csn 3604   class class class wbr 4015   X. cxp 4636   ` cfv 5228  (class class class)co 5888   e. cmpo 5890   RRcr 7823   0cc0 7824   1c1 7825   + caddc 7827   <_ cle 8006   / cdiv 8642   NNcn 8932   2c2 8983   ZZ>=cuz 9541   RR+crp 9666   seqcseq 10458

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942

This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-frec 6405  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552  df-div 8643  df-inn 8933  df-2 8991  df-n0 9190  df-z 9267  df-uz 9542  df-rp 9667  df-seqfrec 10459

This theorem is referenced by:  resqrexlemover  11032  resqrexlemdec  11033  resqrexlemlo  11035  resqrexlemcalc1  11036