Theorem climrecf 39841

Description: A version of climrec 39835 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Hypotheses

Ref	Expression
climrecf.1	|- F/ k ph
climrecf.2	|- F/_ k G
climrecf.3	|- F/_ k H
climrecf.4	|- Z = ( ZZ>= ` M )
climrecf.5	|- ( ph -> M e. ZZ )
climrecf.6	|- ( ph -> G ~~> A )
climrecf.7	|- ( ph -> A =/= 0 )
climrecf.8	|- ( ( ph /\ k e. Z ) -> ( G ` k ) e. ( CC \ { 0 } ) )
climrecf.9	|- ( ( ph /\ k e. Z ) -> ( H ` k ) = ( 1 / ( G ` k ) ) )
climrecf.10	|- ( ph -> H e. W )

Assertion

Ref	Expression
climrecf	|- ( ph -> H ~~> ( 1 / A ) )

Distinct variable group:   k, Z

Allowed substitution hints:   ph( k)   A( k)   G( k)   H( k)   M( k)   W( k)

Proof of Theorem climrecf

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		climrecf.4	. 2 |- Z = ( ZZ>= ` M )
2		climrecf.5	. 2 |- ( ph -> M e. ZZ )
3		climrecf.6	. 2 |- ( ph -> G ~~> A )
4		climrecf.7	. 2 |- ( ph -> A =/= 0 )
5		climrecf.1	. . . . 5 |- F/ k ph
6		nfv 1843	. . . . 5 |- F/ k j e. Z
7	5, 6	nfan 1828	. . . 4 |- F/ k ( ph /\ j e. Z )
8		climrecf.2	. . . . . 6 |- F/_ k G
9		nfcv 2764	. . . . . 6 |- F/_ k j
10	8, 9	nffv 6198	. . . . 5 |- F/_ k ( G ` j )
11	10	nfel1 2779	. . . 4 |- F/ k ( G ` j ) e. ( CC \ { 0 } )
12	7, 11	nfim 1825	. . 3 |- F/ k ( ( ph /\ j e. Z ) -> ( G ` j ) e. ( CC \ { 0 } ) )
13		eleq1 2689	. . . . 5 |- ( k = j -> ( k e. Z <-> j e. Z ) )
14	13	anbi2d 740	. . . 4 |- ( k = j -> ( ( ph /\ k e. Z ) <-> ( ph /\ j e. Z ) ) )
15		fveq2 6191	. . . . 5 |- ( k = j -> ( G ` k ) = ( G ` j ) )
16	15	eleq1d 2686	. . . 4 |- ( k = j -> ( ( G ` k ) e. ( CC \ { 0 } ) <-> ( G ` j ) e. ( CC \ { 0 } ) ) )
17	14, 16	imbi12d 334	. . 3 |- ( k = j -> ( ( ( ph /\ k e. Z ) -> ( G ` k ) e. ( CC \ { 0 } ) ) <-> ( ( ph /\ j e. Z ) -> ( G ` j ) e. ( CC \ { 0 } ) ) ) )
18		climrecf.8	. . 3 |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. ( CC \ { 0 } ) )
19	12, 17, 18	chvar 2262	. 2 |- ( ( ph /\ j e. Z ) -> ( G ` j ) e. ( CC \ { 0 } ) )
20		climrecf.3	. . . . . 6 |- F/_ k H
21	20, 9	nffv 6198	. . . . 5 |- F/_ k ( H ` j )
22		nfcv 2764	. . . . . 6 |- F/_ k 1
23		nfcv 2764	. . . . . 6 |- F/_ k /
24	22, 23, 10	nfov 6676	. . . . 5 |- F/_ k ( 1 / ( G ` j ) )
25	21, 24	nfeq 2776	. . . 4 |- F/ k ( H ` j ) = ( 1 / ( G ` j ) )
26	7, 25	nfim 1825	. . 3 |- F/ k ( ( ph /\ j e. Z ) -> ( H ` j ) = ( 1 / ( G ` j ) ) )
27		fveq2 6191	. . . . 5 |- ( k = j -> ( H ` k ) = ( H ` j ) )
28	15	oveq2d 6666	. . . . 5 |- ( k = j -> ( 1 / ( G ` k ) ) = ( 1 / ( G ` j ) ) )
29	27, 28	eqeq12d 2637	. . . 4 |- ( k = j -> ( ( H ` k ) = ( 1 / ( G ` k ) ) <-> ( H ` j ) = ( 1 / ( G ` j ) ) ) )
30	14, 29	imbi12d 334	. . 3 |- ( k = j -> ( ( ( ph /\ k e. Z ) -> ( H ` k ) = ( 1 / ( G ` k ) ) ) <-> ( ( ph /\ j e. Z ) -> ( H ` j ) = ( 1 / ( G ` j ) ) ) ) )
31		climrecf.9	. . 3 |- ( ( ph /\ k e. Z ) -> ( H ` k ) = ( 1 / ( G ` k ) ) )
32	26, 30, 31	chvar 2262	. 2 |- ( ( ph /\ j e. Z ) -> ( H ` j ) = ( 1 / ( G ` j ) ) )
33		climrecf.10	. 2 |- ( ph -> H e. W )
34	1, 2, 3, 4, 19, 32, 33	climrec 39835	1 |- ( ph -> H ~~> ( 1 / A ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   =/= wne 2794   \ cdif 3571   {csn 4177   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   / cdiv 10684   ZZcz 11377   ZZ>=cuz 11687   ~~> cli 14215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219

This theorem is referenced by:  climdivf  39844