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Theorem mulc1cncfg 27688
Description: A version of mulc1cncf 18927 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 30-Jun-2017.)
Hypotheses
Ref Expression
mulc1cncfg.1  |-  F/_ x F
mulc1cncfg.2  |-  F/ x ph
mulc1cncfg.3  |-  ( ph  ->  F  e.  ( A
-cn-> CC ) )
mulc1cncfg.4  |-  ( ph  ->  B  e.  CC )
Assertion
Ref Expression
mulc1cncfg  |-  ( ph  ->  ( x  e.  A  |->  ( B  x.  ( F `  x )
) )  e.  ( A -cn-> CC ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hints:    ph( x)    F( x)

Proof of Theorem mulc1cncfg
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 mulc1cncfg.4 . . . . . 6  |-  ( ph  ->  B  e.  CC )
2 eqid 2435 . . . . . . 7  |-  ( x  e.  CC  |->  ( B  x.  x ) )  =  ( x  e.  CC  |->  ( B  x.  x ) )
32mulc1cncf 18927 . . . . . 6  |-  ( B  e.  CC  ->  (
x  e.  CC  |->  ( B  x.  x ) )  e.  ( CC
-cn-> CC ) )
41, 3syl 16 . . . . 5  |-  ( ph  ->  ( x  e.  CC  |->  ( B  x.  x
) )  e.  ( CC -cn-> CC ) )
5 cncff 18915 . . . . 5  |-  ( ( x  e.  CC  |->  ( B  x.  x ) )  e.  ( CC
-cn-> CC )  ->  (
x  e.  CC  |->  ( B  x.  x ) ) : CC --> CC )
64, 5syl 16 . . . 4  |-  ( ph  ->  ( x  e.  CC  |->  ( B  x.  x
) ) : CC --> CC )
7 mulc1cncfg.3 . . . . 5  |-  ( ph  ->  F  e.  ( A
-cn-> CC ) )
8 cncff 18915 . . . . 5  |-  ( F  e.  ( A -cn-> CC )  ->  F : A
--> CC )
97, 8syl 16 . . . 4  |-  ( ph  ->  F : A --> CC )
10 fcompt 5896 . . . 4  |-  ( ( ( x  e.  CC  |->  ( B  x.  x
) ) : CC --> CC  /\  F : A --> CC )  ->  ( ( x  e.  CC  |->  ( B  x.  x ) )  o.  F )  =  ( t  e.  A  |->  ( ( x  e.  CC  |->  ( B  x.  x ) ) `
 ( F `  t ) ) ) )
116, 9, 10syl2anc 643 . . 3  |-  ( ph  ->  ( ( x  e.  CC  |->  ( B  x.  x ) )  o.  F )  =  ( t  e.  A  |->  ( ( x  e.  CC  |->  ( B  x.  x
) ) `  ( F `  t )
) ) )
129fnvinran 27652 . . . . . 6  |-  ( (
ph  /\  t  e.  A )  ->  ( F `  t )  e.  CC )
131adantr 452 . . . . . . 7  |-  ( (
ph  /\  t  e.  A )  ->  B  e.  CC )
1413, 12mulcld 9100 . . . . . 6  |-  ( (
ph  /\  t  e.  A )  ->  ( B  x.  ( F `  t ) )  e.  CC )
15 mulc1cncfg.1 . . . . . . . 8  |-  F/_ x F
16 nfcv 2571 . . . . . . . 8  |-  F/_ x
t
1715, 16nffv 5727 . . . . . . 7  |-  F/_ x
( F `  t
)
18 nfcv 2571 . . . . . . . 8  |-  F/_ x B
19 nfcv 2571 . . . . . . . 8  |-  F/_ x  x.
2018, 19, 17nfov 6096 . . . . . . 7  |-  F/_ x
( B  x.  ( F `  t )
)
21 oveq2 6081 . . . . . . 7  |-  ( x  =  ( F `  t )  ->  ( B  x.  x )  =  ( B  x.  ( F `  t ) ) )
2217, 20, 21, 2fvmptf 5813 . . . . . 6  |-  ( ( ( F `  t
)  e.  CC  /\  ( B  x.  ( F `  t )
)  e.  CC )  ->  ( ( x  e.  CC  |->  ( B  x.  x ) ) `
 ( F `  t ) )  =  ( B  x.  ( F `  t )
) )
2312, 14, 22syl2anc 643 . . . . 5  |-  ( (
ph  /\  t  e.  A )  ->  (
( x  e.  CC  |->  ( B  x.  x
) ) `  ( F `  t )
)  =  ( B  x.  ( F `  t ) ) )
2423mpteq2dva 4287 . . . 4  |-  ( ph  ->  ( t  e.  A  |->  ( ( x  e.  CC  |->  ( B  x.  x ) ) `  ( F `  t ) ) )  =  ( t  e.  A  |->  ( B  x.  ( F `
 t ) ) ) )
25 nfcv 2571 . . . . . 6  |-  F/_ t B
26 nfcv 2571 . . . . . 6  |-  F/_ t  x.
27 nfcv 2571 . . . . . 6  |-  F/_ t
( F `  x
)
2825, 26, 27nfov 6096 . . . . 5  |-  F/_ t
( B  x.  ( F `  x )
)
29 fveq2 5720 . . . . . 6  |-  ( t  =  x  ->  ( F `  t )  =  ( F `  x ) )
3029oveq2d 6089 . . . . 5  |-  ( t  =  x  ->  ( B  x.  ( F `  t ) )  =  ( B  x.  ( F `  x )
) )
3120, 28, 30cbvmpt 4291 . . . 4  |-  ( t  e.  A  |->  ( B  x.  ( F `  t ) ) )  =  ( x  e.  A  |->  ( B  x.  ( F `  x ) ) )
3224, 31syl6eq 2483 . . 3  |-  ( ph  ->  ( t  e.  A  |->  ( ( x  e.  CC  |->  ( B  x.  x ) ) `  ( F `  t ) ) )  =  ( x  e.  A  |->  ( B  x.  ( F `
 x ) ) ) )
3311, 32eqtrd 2467 . 2  |-  ( ph  ->  ( ( x  e.  CC  |->  ( B  x.  x ) )  o.  F )  =  ( x  e.  A  |->  ( B  x.  ( F `
 x ) ) ) )
347, 4cncfco 18929 . 2  |-  ( ph  ->  ( ( x  e.  CC  |->  ( B  x.  x ) )  o.  F )  e.  ( A -cn-> CC ) )
3533, 34eqeltrrd 2510 1  |-  ( ph  ->  ( x  e.  A  |->  ( B  x.  ( F `  x )
) )  e.  ( A -cn-> CC ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   F/wnf 1553    = wceq 1652    e. wcel 1725   F/_wnfc 2558    e. cmpt 4258    o. ccom 4874   -->wf 5442   ` cfv 5446  (class class class)co 6073   CCcc 8980    x. cmul 8987   -cn->ccncf 18898
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-cncf 18900
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