Theorem resqrexlemfp1 11569

Description: Lemma for resqrex 11586. 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 9792	. . . . . 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 9792	. . . . . 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 11565	. . . . . 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 11566	. . . . 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 10726	. . 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 5640	. . 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 5640	. . . 4 |- ( F ` N ) = ( seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) ) ` N )
16	15	oveq1i 6027	. . 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 2289	. 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 6025	. . . . . . 7 |- ( y = c -> ( A / y ) = ( A / c ) )
20	18, 19	oveq12d 6035	. . . . . 6 |- ( y = c -> ( y + ( A / y ) ) = ( c + ( A / c ) ) )
21	20	oveq1d 6032	. . . . 5 |- ( y = c -> ( ( y + ( A / y ) ) / 2 ) = ( ( c + ( A / c ) ) / 2 ) )
22		eqidd 2232	. . . . 5 |- ( z = d -> ( ( c + ( A / c ) ) / 2 ) = ( ( c + ( A / c ) ) / 2 ) )
23	21, 22	cbvmpov 6100	. . . 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 6025	. . . . . 6 |- ( c = ( F ` N ) -> ( A / c ) = ( A / ( F ` N ) ) )
27	25, 26	oveq12d 6035	. . . . 5 |- ( c = ( F ` N ) -> ( c + ( A / c ) ) = ( ( F ` N ) + ( A / ( F ` N ) ) ) )
28	27	oveq1d 6032	. . . 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 11567	. . . 4 |- ( ph -> F : NN --> RR+ )
31	30	ffvelcdmda 5782	. . 3 |- ( ( ph /\ N e. NN ) -> ( F ` N ) e. RR+ )
32		peano2nn 9154	. . . 4 |- ( N e. NN -> ( N + 1 ) e. NN )
33	5, 6	resqrexlem1arp 11565	. . . 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 9930	. . . . 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 9962	. . . . 5 |- ( ( ph /\ N e. NN ) -> ( A / ( F ` N ) ) e. RR )
38	35, 37	readdcld 8208	. . . 4 |- ( ( ph /\ N e. NN ) -> ( ( F ` N ) + ( A / ( F ` N ) ) ) e. RR )
39	38	rehalfcld 9390	. . 3 |- ( ( ph /\ N e. NN ) -> ( ( ( F ` N ) + ( A / ( F ` N ) ) ) / 2 ) e. RR )
40	24, 29, 31, 34, 39	ovmpod 6148	. 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 2264	1 |- ( ( ph /\ N e. NN ) -> ( F ` ( N + 1 ) ) = ( ( ( F ` N ) + ( A / ( F ` N ) ) ) / 2 ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   {csn 3669   class class class wbr 4088   X. cxp 4723   ` cfv 5326  (class class class)co 6017   e. cmpo 6019   RRcr 8030   0cc0 8031   1c1 8032   + caddc 8034   <_ cle 8214   / cdiv 8851   NNcn 9142   2c2 9193   ZZ>=cuz 9754   RR+crp 9887   seqcseq 10708

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-n0 9402  df-z 9479  df-uz 9755  df-rp 9888  df-seqfrec 10709

This theorem is referenced by:  resqrexlemover  11570  resqrexlemdec  11571  resqrexlemlo  11573  resqrexlemcalc1  11574