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Theorem mulc1cncfg 27824
Description: A version of mulc1cncf 18425 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 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
2 eqid 2296 . . . . . . . 8  |-  ( x  e.  CC  |->  ( B  x.  x ) )  =  ( x  e.  CC  |->  ( B  x.  x ) )
32mulc1cncf 18425 . . . . . . 7  |-  ( B  e.  CC  ->  (
x  e.  CC  |->  ( B  x.  x ) )  e.  ( CC
-cn-> CC ) )
41, 3syl 15 . . . . . 6  |-  ( ph  ->  ( x  e.  CC  |->  ( B  x.  x
) )  e.  ( CC -cn-> CC ) )
5 cncff 18413 . . . . . 6  |-  ( ( x  e.  CC  |->  ( B  x.  x ) )  e.  ( CC
-cn-> CC )  ->  (
x  e.  CC  |->  ( B  x.  x ) ) : CC --> CC )
64, 5syl 15 . . . . 5  |-  ( ph  ->  ( x  e.  CC  |->  ( B  x.  x
) ) : CC --> CC )
7 mulc1cncfg.3 . . . . . 6  |-  ( ph  ->  F  e.  ( A
-cn-> CC ) )
8 cncff 18413 . . . . . 6  |-  ( F  e.  ( A -cn-> CC )  ->  F : A
--> CC )
97, 8syl 15 . . . . 5  |-  ( ph  ->  F : A --> CC )
106, 9jca 518 . . . 4  |-  ( ph  ->  ( ( x  e.  CC  |->  ( B  x.  x ) ) : CC --> CC  /\  F : A --> CC ) )
11 fcompt 5710 . . . 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 ) ) ) )
1210, 11syl 15 . . 3  |-  ( ph  ->  ( ( x  e.  CC  |->  ( B  x.  x ) )  o.  F )  =  ( t  e.  A  |->  ( ( x  e.  CC  |->  ( B  x.  x
) ) `  ( F `  t )
) ) )
139adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  A )  ->  F : A --> CC )
14 simpr 447 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  A )  ->  t  e.  A )
1513, 14jca 518 . . . . . . . 8  |-  ( (
ph  /\  t  e.  A )  ->  ( F : A --> CC  /\  t  e.  A )
)
16 ffvelrn 5679 . . . . . . . 8  |-  ( ( F : A --> CC  /\  t  e.  A )  ->  ( F `  t
)  e.  CC )
1715, 16syl 15 . . . . . . 7  |-  ( (
ph  /\  t  e.  A )  ->  ( F `  t )  e.  CC )
181adantr 451 . . . . . . . 8  |-  ( (
ph  /\  t  e.  A )  ->  B  e.  CC )
1918, 17mulcld 8871 . . . . . . 7  |-  ( (
ph  /\  t  e.  A )  ->  ( B  x.  ( F `  t ) )  e.  CC )
2017, 19jca 518 . . . . . 6  |-  ( (
ph  /\  t  e.  A )  ->  (
( F `  t
)  e.  CC  /\  ( B  x.  ( F `  t )
)  e.  CC ) )
21 mulc1cncfg.1 . . . . . . . 8  |-  F/_ x F
22 nfcv 2432 . . . . . . . 8  |-  F/_ x
t
2321, 22nffv 5548 . . . . . . 7  |-  F/_ x
( F `  t
)
24 nfcv 2432 . . . . . . . 8  |-  F/_ x B
25 nfcv 2432 . . . . . . . 8  |-  F/_ x  x.
2624, 25, 23nfov 5897 . . . . . . 7  |-  F/_ x
( B  x.  ( F `  t )
)
27 oveq2 5882 . . . . . . 7  |-  ( x  =  ( F `  t )  ->  ( B  x.  x )  =  ( B  x.  ( F `  t ) ) )
2823, 26, 27, 2fvmptf 5632 . . . . . 6  |-  ( ( ( F `  t
)  e.  CC  /\  ( B  x.  ( F `  t )
)  e.  CC )  ->  ( ( x  e.  CC  |->  ( B  x.  x ) ) `
 ( F `  t ) )  =  ( B  x.  ( F `  t )
) )
2920, 28syl 15 . . . . 5  |-  ( (
ph  /\  t  e.  A )  ->  (
( x  e.  CC  |->  ( B  x.  x
) ) `  ( F `  t )
)  =  ( B  x.  ( F `  t ) ) )
3029mpteq2dva 4122 . . . 4  |-  ( ph  ->  ( t  e.  A  |->  ( ( x  e.  CC  |->  ( B  x.  x ) ) `  ( F `  t ) ) )  =  ( t  e.  A  |->  ( B  x.  ( F `
 t ) ) ) )
31 nfcv 2432 . . . . . 6  |-  F/_ t B
32 nfcv 2432 . . . . . 6  |-  F/_ t  x.
33 nfcv 2432 . . . . . 6  |-  F/_ t
( F `  x
)
3431, 32, 33nfov 5897 . . . . 5  |-  F/_ t
( B  x.  ( F `  x )
)
35 fveq2 5541 . . . . . 6  |-  ( t  =  x  ->  ( F `  t )  =  ( F `  x ) )
3635oveq2d 5890 . . . . 5  |-  ( t  =  x  ->  ( B  x.  ( F `  t ) )  =  ( B  x.  ( F `  x )
) )
3726, 34, 36cbvmpt 4126 . . . 4  |-  ( t  e.  A  |->  ( B  x.  ( F `  t ) ) )  =  ( x  e.  A  |->  ( B  x.  ( F `  x ) ) )
3830, 37syl6eq 2344 . . 3  |-  ( ph  ->  ( t  e.  A  |->  ( ( x  e.  CC  |->  ( B  x.  x ) ) `  ( F `  t ) ) )  =  ( x  e.  A  |->  ( B  x.  ( F `
 x ) ) ) )
3912, 38eqtrd 2328 . 2  |-  ( ph  ->  ( ( x  e.  CC  |->  ( B  x.  x ) )  o.  F )  =  ( x  e.  A  |->  ( B  x.  ( F `
 x ) ) ) )
407, 4cncfco 18427 . . 3  |-  ( ph  ->  ( ( x  e.  CC  |->  ( B  x.  x ) )  o.  F )  e.  ( A -cn-> CC ) )
4140idi 2 . 2  |-  ( ph  ->  ( ( x  e.  CC  |->  ( B  x.  x ) )  o.  F )  e.  ( A -cn-> CC ) )
4239, 41eqeltrrd 2371 1  |-  ( ph  ->  ( x  e.  A  |->  ( B  x.  ( F `  x )
) )  e.  ( A -cn-> CC ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   F/wnf 1534    = wceq 1632    e. wcel 1696   F/_wnfc 2419    e. cmpt 4093    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751    x. cmul 8758   -cn->ccncf 18396
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-cncf 18398
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