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Theorem mulc1cncfg 31822
Description: A version of mulc1cncf 21575 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 2454 . . . . . . 7  |-  ( x  e.  CC  |->  ( B  x.  x ) )  =  ( x  e.  CC  |->  ( B  x.  x ) )
32mulc1cncf 21575 . . . . . 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 21563 . . . . 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 21563 . . . . 5  |-  ( F  e.  ( A -cn-> CC )  ->  F : A
--> CC )
97, 8syl 16 . . . 4  |-  ( ph  ->  F : A --> CC )
10 fcompt 6043 . . . 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 659 . . 3  |-  ( ph  ->  ( ( x  e.  CC  |->  ( B  x.  x ) )  o.  F )  =  ( t  e.  A  |->  ( ( x  e.  CC  |->  ( B  x.  x
) ) `  ( F `  t )
) ) )
129fnvinran 31629 . . . . . 6  |-  ( (
ph  /\  t  e.  A )  ->  ( F `  t )  e.  CC )
131adantr 463 . . . . . . 7  |-  ( (
ph  /\  t  e.  A )  ->  B  e.  CC )
1413, 12mulcld 9605 . . . . . 6  |-  ( (
ph  /\  t  e.  A )  ->  ( B  x.  ( F `  t ) )  e.  CC )
15 mulc1cncfg.1 . . . . . . . 8  |-  F/_ x F
16 nfcv 2616 . . . . . . . 8  |-  F/_ x
t
1715, 16nffv 5855 . . . . . . 7  |-  F/_ x
( F `  t
)
18 nfcv 2616 . . . . . . . 8  |-  F/_ x B
19 nfcv 2616 . . . . . . . 8  |-  F/_ x  x.
2018, 19, 17nfov 6296 . . . . . . 7  |-  F/_ x
( B  x.  ( F `  t )
)
21 oveq2 6278 . . . . . . 7  |-  ( x  =  ( F `  t )  ->  ( B  x.  x )  =  ( B  x.  ( F `  t ) ) )
2217, 20, 21, 2fvmptf 5948 . . . . . 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 659 . . . . 5  |-  ( (
ph  /\  t  e.  A )  ->  (
( x  e.  CC  |->  ( B  x.  x
) ) `  ( F `  t )
)  =  ( B  x.  ( F `  t ) ) )
2423mpteq2dva 4525 . . . 4  |-  ( ph  ->  ( t  e.  A  |->  ( ( x  e.  CC  |->  ( B  x.  x ) ) `  ( F `  t ) ) )  =  ( t  e.  A  |->  ( B  x.  ( F `
 t ) ) ) )
25 nfcv 2616 . . . . . 6  |-  F/_ t B
26 nfcv 2616 . . . . . 6  |-  F/_ t  x.
27 nfcv 2616 . . . . . 6  |-  F/_ t
( F `  x
)
2825, 26, 27nfov 6296 . . . . 5  |-  F/_ t
( B  x.  ( F `  x )
)
29 fveq2 5848 . . . . . 6  |-  ( t  =  x  ->  ( F `  t )  =  ( F `  x ) )
3029oveq2d 6286 . . . . 5  |-  ( t  =  x  ->  ( B  x.  ( F `  t ) )  =  ( B  x.  ( F `  x )
) )
3120, 28, 30cbvmpt 4529 . . . 4  |-  ( t  e.  A  |->  ( B  x.  ( F `  t ) ) )  =  ( x  e.  A  |->  ( B  x.  ( F `  x ) ) )
3224, 31syl6eq 2511 . . 3  |-  ( ph  ->  ( t  e.  A  |->  ( ( x  e.  CC  |->  ( B  x.  x ) ) `  ( F `  t ) ) )  =  ( x  e.  A  |->  ( B  x.  ( F `
 x ) ) ) )
3311, 32eqtrd 2495 . 2  |-  ( ph  ->  ( ( x  e.  CC  |->  ( B  x.  x ) )  o.  F )  =  ( x  e.  A  |->  ( B  x.  ( F `
 x ) ) ) )
347, 4cncfco 21577 . 2  |-  ( ph  ->  ( ( x  e.  CC  |->  ( B  x.  x ) )  o.  F )  e.  ( A -cn-> CC ) )
3533, 34eqeltrrd 2543 1  |-  ( ph  ->  ( x  e.  A  |->  ( B  x.  ( F `  x )
) )  e.  ( A -cn-> CC ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398   F/wnf 1621    e. wcel 1823   F/_wnfc 2602    |-> cmpt 4497    o. ccom 4992   -->wf 5566   ` cfv 5570  (class class class)co 6270   CCcc 9479    x. cmul 9486   -cn->ccncf 21546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-seq 12090  df-exp 12149  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-cncf 21548
This theorem is referenced by: (None)
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