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Theorem mulc1cncfg 37243
Description: A version of mulc1cncf 21826 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 2420 . . . . . . 7  |-  ( x  e.  CC  |->  ( B  x.  x ) )  =  ( x  e.  CC  |->  ( B  x.  x ) )
32mulc1cncf 21826 . . . . . 6  |-  ( B  e.  CC  ->  (
x  e.  CC  |->  ( B  x.  x ) )  e.  ( CC
-cn-> CC ) )
41, 3syl 17 . . . . 5  |-  ( ph  ->  ( x  e.  CC  |->  ( B  x.  x
) )  e.  ( CC -cn-> CC ) )
5 cncff 21814 . . . . 5  |-  ( ( x  e.  CC  |->  ( B  x.  x ) )  e.  ( CC
-cn-> CC )  ->  (
x  e.  CC  |->  ( B  x.  x ) ) : CC --> CC )
64, 5syl 17 . . . 4  |-  ( ph  ->  ( x  e.  CC  |->  ( B  x.  x
) ) : CC --> CC )
7 mulc1cncfg.3 . . . . 5  |-  ( ph  ->  F  e.  ( A
-cn-> CC ) )
8 cncff 21814 . . . . 5  |-  ( F  e.  ( A -cn-> CC )  ->  F : A
--> CC )
97, 8syl 17 . . . 4  |-  ( ph  ->  F : A --> CC )
10 fcompt 6065 . . . 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 665 . . 3  |-  ( ph  ->  ( ( x  e.  CC  |->  ( B  x.  x ) )  o.  F )  =  ( t  e.  A  |->  ( ( x  e.  CC  |->  ( B  x.  x
) ) `  ( F `  t )
) ) )
129fnvinran 36979 . . . . . 6  |-  ( (
ph  /\  t  e.  A )  ->  ( F `  t )  e.  CC )
131adantr 466 . . . . . . 7  |-  ( (
ph  /\  t  e.  A )  ->  B  e.  CC )
1413, 12mulcld 9652 . . . . . 6  |-  ( (
ph  /\  t  e.  A )  ->  ( B  x.  ( F `  t ) )  e.  CC )
15 mulc1cncfg.1 . . . . . . . 8  |-  F/_ x F
16 nfcv 2582 . . . . . . . 8  |-  F/_ x
t
1715, 16nffv 5879 . . . . . . 7  |-  F/_ x
( F `  t
)
18 nfcv 2582 . . . . . . . 8  |-  F/_ x B
19 nfcv 2582 . . . . . . . 8  |-  F/_ x  x.
2018, 19, 17nfov 6322 . . . . . . 7  |-  F/_ x
( B  x.  ( F `  t )
)
21 oveq2 6304 . . . . . . 7  |-  ( x  =  ( F `  t )  ->  ( B  x.  x )  =  ( B  x.  ( F `  t ) ) )
2217, 20, 21, 2fvmptf 5973 . . . . . 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 665 . . . . 5  |-  ( (
ph  /\  t  e.  A )  ->  (
( x  e.  CC  |->  ( B  x.  x
) ) `  ( F `  t )
)  =  ( B  x.  ( F `  t ) ) )
2423mpteq2dva 4503 . . . 4  |-  ( ph  ->  ( t  e.  A  |->  ( ( x  e.  CC  |->  ( B  x.  x ) ) `  ( F `  t ) ) )  =  ( t  e.  A  |->  ( B  x.  ( F `
 t ) ) ) )
25 nfcv 2582 . . . . . 6  |-  F/_ t B
26 nfcv 2582 . . . . . 6  |-  F/_ t  x.
27 nfcv 2582 . . . . . 6  |-  F/_ t
( F `  x
)
2825, 26, 27nfov 6322 . . . . 5  |-  F/_ t
( B  x.  ( F `  x )
)
29 fveq2 5872 . . . . . 6  |-  ( t  =  x  ->  ( F `  t )  =  ( F `  x ) )
3029oveq2d 6312 . . . . 5  |-  ( t  =  x  ->  ( B  x.  ( F `  t ) )  =  ( B  x.  ( F `  x )
) )
3120, 28, 30cbvmpt 4508 . . . 4  |-  ( t  e.  A  |->  ( B  x.  ( F `  t ) ) )  =  ( x  e.  A  |->  ( B  x.  ( F `  x ) ) )
3224, 31syl6eq 2477 . . 3  |-  ( ph  ->  ( t  e.  A  |->  ( ( x  e.  CC  |->  ( B  x.  x ) ) `  ( F `  t ) ) )  =  ( x  e.  A  |->  ( B  x.  ( F `
 x ) ) ) )
3311, 32eqtrd 2461 . 2  |-  ( ph  ->  ( ( x  e.  CC  |->  ( B  x.  x ) )  o.  F )  =  ( x  e.  A  |->  ( B  x.  ( F `
 x ) ) ) )
347, 4cncfco 21828 . 2  |-  ( ph  ->  ( ( x  e.  CC  |->  ( B  x.  x ) )  o.  F )  e.  ( A -cn-> CC ) )
3533, 34eqeltrrd 2509 1  |-  ( ph  ->  ( x  e.  A  |->  ( B  x.  ( F `  x )
) )  e.  ( A -cn-> CC ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437   F/wnf 1663    e. wcel 1867   F/_wnfc 2568    |-> cmpt 4475    o. ccom 4849   -->wf 5588   ` cfv 5592  (class class class)co 6296   CCcc 9526    x. cmul 9533   -cn->ccncf 21797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-er 7362  df-map 7473  df-en 7569  df-dom 7570  df-sdom 7571  df-sup 7953  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-n0 10859  df-z 10927  df-uz 11149  df-rp 11292  df-seq 12200  df-exp 12259  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-cncf 21799
This theorem is referenced by: (None)
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