Theorem resqrexlemfp1 11435

Description: Lemma for resqrex 11452. 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 9720	. . . . . 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 9720	. . . . . 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 11431	. . . . . 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 11432	. . . . 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 10647	. . 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 5600	. . 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 5600	. . . 4 |- ( F ` N ) = ( seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) ) ` N )
16	15	oveq1i 5977	. . 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 2265	. 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 5975	. . . . . . 7 |- ( y = c -> ( A / y ) = ( A / c ) )
20	18, 19	oveq12d 5985	. . . . . 6 |- ( y = c -> ( y + ( A / y ) ) = ( c + ( A / c ) ) )
21	20	oveq1d 5982	. . . . 5 |- ( y = c -> ( ( y + ( A / y ) ) / 2 ) = ( ( c + ( A / c ) ) / 2 ) )
22		eqidd 2208	. . . . 5 |- ( z = d -> ( ( c + ( A / c ) ) / 2 ) = ( ( c + ( A / c ) ) / 2 ) )
23	21, 22	cbvmpov 6048	. . . 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 5975	. . . . . 6 |- ( c = ( F ` N ) -> ( A / c ) = ( A / ( F ` N ) ) )
27	25, 26	oveq12d 5985	. . . . 5 |- ( c = ( F ` N ) -> ( c + ( A / c ) ) = ( ( F ` N ) + ( A / ( F ` N ) ) ) )
28	27	oveq1d 5982	. . . 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 11433	. . . 4 |- ( ph -> F : NN --> RR+ )
31	30	ffvelcdmda 5738	. . 3 |- ( ( ph /\ N e. NN ) -> ( F ` N ) e. RR+ )
32		peano2nn 9083	. . . 4 |- ( N e. NN -> ( N + 1 ) e. NN )
33	5, 6	resqrexlem1arp 11431	. . . 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 9853	. . . . 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 9885	. . . . 5 |- ( ( ph /\ N e. NN ) -> ( A / ( F ` N ) ) e. RR )
38	35, 37	readdcld 8137	. . . 4 |- ( ( ph /\ N e. NN ) -> ( ( F ` N ) + ( A / ( F ` N ) ) ) e. RR )
39	38	rehalfcld 9319	. . 3 |- ( ( ph /\ N e. NN ) -> ( ( ( F ` N ) + ( A / ( F ` N ) ) ) / 2 ) e. RR )
40	24, 29, 31, 34, 39	ovmpod 6096	. 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 2240	1 |- ( ( ph /\ N e. NN ) -> ( F ` ( N + 1 ) ) = ( ( ( F ` N ) + ( A / ( F ` N ) ) ) / 2 ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   {csn 3643   class class class wbr 4059   X. cxp 4691   ` cfv 5290  (class class class)co 5967   e. cmpo 5969   RRcr 7959   0cc0 7960   1c1 7961   + caddc 7963   <_ cle 8143   / cdiv 8780   NNcn 9071   2c2 9122   ZZ>=cuz 9683   RR+crp 9810   seqcseq 10629

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-n0 9331  df-z 9408  df-uz 9684  df-rp 9811  df-seqfrec 10630

This theorem is referenced by:  resqrexlemover  11436  resqrexlemdec  11437  resqrexlemlo  11439  resqrexlemcalc1  11440