Theorem resqrexlemfp1 11035

Description: Lemma for resqrex 11052. 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 9581	. . . . . 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 9581	. . . . . 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 11031	. . . . . 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 11032	. . . . 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 10479	. . 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 5530	. . 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 5530	. . . 4 |- ( F ` N ) = ( seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) ) ` N )
16	15	oveq1i 5900	. . 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 2246	. 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 5898	. . . . . . 7 |- ( y = c -> ( A / y ) = ( A / c ) )
20	18, 19	oveq12d 5908	. . . . . 6 |- ( y = c -> ( y + ( A / y ) ) = ( c + ( A / c ) ) )
21	20	oveq1d 5905	. . . . 5 |- ( y = c -> ( ( y + ( A / y ) ) / 2 ) = ( ( c + ( A / c ) ) / 2 ) )
22		eqidd 2189	. . . . 5 |- ( z = d -> ( ( c + ( A / c ) ) / 2 ) = ( ( c + ( A / c ) ) / 2 ) )
23	21, 22	cbvmpov 5970	. . . 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 5898	. . . . . 6 |- ( c = ( F ` N ) -> ( A / c ) = ( A / ( F ` N ) ) )
27	25, 26	oveq12d 5908	. . . . 5 |- ( c = ( F ` N ) -> ( c + ( A / c ) ) = ( ( F ` N ) + ( A / ( F ` N ) ) ) )
28	27	oveq1d 5905	. . . 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 11033	. . . 4 |- ( ph -> F : NN --> RR+ )
31	30	ffvelcdmda 5666	. . 3 |- ( ( ph /\ N e. NN ) -> ( F ` N ) e. RR+ )
32		peano2nn 8948	. . . 4 |- ( N e. NN -> ( N + 1 ) e. NN )
33	5, 6	resqrexlem1arp 11031	. . . 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 9713	. . . . 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 9745	. . . . 5 |- ( ( ph /\ N e. NN ) -> ( A / ( F ` N ) ) e. RR )
38	35, 37	readdcld 8004	. . . 4 |- ( ( ph /\ N e. NN ) -> ( ( F ` N ) + ( A / ( F ` N ) ) ) e. RR )
39	38	rehalfcld 9182	. . 3 |- ( ( ph /\ N e. NN ) -> ( ( ( F ` N ) + ( A / ( F ` N ) ) ) / 2 ) e. RR )
40	24, 29, 31, 34, 39	ovmpod 6018	. 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 2221	1 |- ( ( ph /\ N e. NN ) -> ( F ` ( N + 1 ) ) = ( ( ( F ` N ) + ( A / ( F ` N ) ) ) / 2 ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1363   e. wcel 2159   {csn 3606   class class class wbr 4017   X. cxp 4638   ` cfv 5230  (class class class)co 5890   e. cmpo 5892   RRcr 7827   0cc0 7828   1c1 7829   + caddc 7831   <_ cle 8010   / cdiv 8646   NNcn 8936   2c2 8987   ZZ>=cuz 9545   RR+crp 9670   seqcseq 10462

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 2161  ax-14 2162  ax-ext 2170  ax-coll 4132  ax-sep 4135  ax-nul 4143  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-iinf 4601  ax-cnex 7919  ax-resscn 7920  ax-1cn 7921  ax-1re 7922  ax-icn 7923  ax-addcl 7924  ax-addrcl 7925  ax-mulcl 7926  ax-mulrcl 7927  ax-addcom 7928  ax-mulcom 7929  ax-addass 7930  ax-mulass 7931  ax-distr 7932  ax-i2m1 7933  ax-0lt1 7934  ax-1rid 7935  ax-0id 7936  ax-rnegex 7937  ax-precex 7938  ax-cnre 7939  ax-pre-ltirr 7940  ax-pre-ltwlin 7941  ax-pre-lttrn 7942  ax-pre-apti 7943  ax-pre-ltadd 7944  ax-pre-mulgt0 7945  ax-pre-mulext 7946

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 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-nel 2455  df-ral 2472  df-rex 2473  df-reu 2474  df-rmo 2475  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-nul 3437  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-tr 4116  df-id 4307  df-po 4310  df-iso 4311  df-iord 4380  df-on 4382  df-ilim 4383  df-suc 4385  df-iom 4604  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-riota 5846  df-ov 5893  df-oprab 5894  df-mpo 5895  df-1st 6158  df-2nd 6159  df-recs 6323  df-frec 6409  df-pnf 8011  df-mnf 8012  df-xr 8013  df-ltxr 8014  df-le 8015  df-sub 8147  df-neg 8148  df-reap 8549  df-ap 8556  df-div 8647  df-inn 8937  df-2 8995  df-n0 9194  df-z 9271  df-uz 9546  df-rp 9671  df-seqfrec 10463

This theorem is referenced by:  resqrexlemover  11036  resqrexlemdec  11037  resqrexlemlo  11039  resqrexlemcalc1  11040