Theorem mulc1cncfg 39821

Description: A version of mulc1cncf 22708 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.

Step	Hyp	Ref	Expression
1		mulc1cncfg.4	. . . . . 6 |- ( ph -> B e. CC )
2		eqid 2622	. . . . . . 7 |- ( x e. CC |-> ( B x. x ) ) = ( x e. CC |-> ( B x. x ) )
3	2	mulc1cncf 22708	. . . . . 6 |- ( B e. CC -> ( x e. CC |-> ( B x. x ) ) e. ( CC -cn-> CC ) )
4	1, 3	syl 17	. . . . 5 |- ( ph -> ( x e. CC |-> ( B x. x ) ) e. ( CC -cn-> CC ) )
5		cncff 22696	. . . . 5 |- ( ( x e. CC |-> ( B x. x ) ) e. ( CC -cn-> CC ) -> ( x e. CC |-> ( B x. x ) ) : CC --> CC )
6	4, 5	syl 17	. . . 4 |- ( ph -> ( x e. CC |-> ( B x. x ) ) : CC --> CC )
7		mulc1cncfg.3	. . . . 5 |- ( ph -> F e. ( A -cn-> CC ) )
8		cncff 22696	. . . . 5 |- ( F e. ( A -cn-> CC ) -> F : A --> CC )
9	7, 8	syl 17	. . . 4 |- ( ph -> F : A --> CC )
10		fcompt 6400	. . . 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 ) ) ) )
11	6, 9, 10	syl2anc 693	. . 3 |- ( ph -> ( ( x e. CC |-> ( B x. x ) ) o. F ) = ( t e. A |-> ( ( x e. CC |-> ( B x. x ) ) ` ( F ` t ) ) ) )
12	9	ffvelrnda 6359	. . . . . 6 |- ( ( ph /\ t e. A ) -> ( F ` t ) e. CC )
13	1	adantr 481	. . . . . . 7 |- ( ( ph /\ t e. A ) -> B e. CC )
14	13, 12	mulcld 10060	. . . . . 6 |- ( ( ph /\ t e. A ) -> ( B x. ( F ` t ) ) e. CC )
15		mulc1cncfg.1	. . . . . . . 8 |- F/_ x F
16		nfcv 2764	. . . . . . . 8 |- F/_ x t
17	15, 16	nffv 6198	. . . . . . 7 |- F/_ x ( F ` t )
18		nfcv 2764	. . . . . . . 8 |- F/_ x B
19		nfcv 2764	. . . . . . . 8 |- F/_ x x.
20	18, 19, 17	nfov 6676	. . . . . . 7 |- F/_ x ( B x. ( F ` t ) )
21		oveq2 6658	. . . . . . 7 |- ( x = ( F ` t ) -> ( B x. x ) = ( B x. ( F ` t ) ) )
22	17, 20, 21, 2	fvmptf 6301	. . . . . 6 |- ( ( ( F ` t ) e. CC /\ ( B x. ( F ` t ) ) e. CC ) -> ( ( x e. CC |-> ( B x. x ) ) ` ( F ` t ) ) = ( B x. ( F ` t ) ) )
23	12, 14, 22	syl2anc 693	. . . . 5 |- ( ( ph /\ t e. A ) -> ( ( x e. CC |-> ( B x. x ) ) ` ( F ` t ) ) = ( B x. ( F ` t ) ) )
24	23	mpteq2dva 4744	. . . 4 |- ( ph -> ( t e. A |-> ( ( x e. CC |-> ( B x. x ) ) ` ( F ` t ) ) ) = ( t e. A |-> ( B x. ( F ` t ) ) ) )
25		nfcv 2764	. . . . . 6 |- F/_ t B
26		nfcv 2764	. . . . . 6 |- F/_ t x.
27		nfcv 2764	. . . . . 6 |- F/_ t ( F ` x )
28	25, 26, 27	nfov 6676	. . . . 5 |- F/_ t ( B x. ( F ` x ) )
29		fveq2 6191	. . . . . 6 |- ( t = x -> ( F ` t ) = ( F ` x ) )
30	29	oveq2d 6666	. . . . 5 |- ( t = x -> ( B x. ( F ` t ) ) = ( B x. ( F ` x ) ) )
31	20, 28, 30	cbvmpt 4749	. . . 4 |- ( t e. A |-> ( B x. ( F ` t ) ) ) = ( x e. A |-> ( B x. ( F ` x ) ) )
32	24, 31	syl6eq 2672	. . 3 |- ( ph -> ( t e. A |-> ( ( x e. CC |-> ( B x. x ) ) ` ( F ` t ) ) ) = ( x e. A |-> ( B x. ( F ` x ) ) ) )
33	11, 32	eqtrd 2656	. 2 |- ( ph -> ( ( x e. CC |-> ( B x. x ) ) o. F ) = ( x e. A |-> ( B x. ( F ` x ) ) ) )
34	7, 4	cncfco 22710	. 2 |- ( ph -> ( ( x e. CC |-> ( B x. x ) ) o. F ) e. ( A -cn-> CC ) )
35	33, 34	eqeltrrd 2702	1 |- ( ph -> ( x e. A |-> ( B x. ( F ` x ) ) ) e. ( A -cn-> CC ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   |-> cmpt 4729   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   x. cmul 9941   -cn->ccncf 22679

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-map 7859  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-cncf 22681

This theorem is referenced by: (None)