Theorem rmulccn 29974

Description: Multiplication by a real constant is a continuous function. (Contributed by Thierry Arnoux, 23-May-2017.)

Hypotheses

Ref	Expression
rmulccn.1	|- J = ( topGen ` ran (,) )
rmulccn.2	|- ( ph -> C e. RR )

Assertion

Ref	Expression
rmulccn	|- ( ph -> ( x e. RR |-> ( x x. C ) ) e. ( J Cn J ) )

Distinct variable groups:   x, C   ph, x

Allowed substitution hint:   J( x)

Proof of Theorem rmulccn

Dummy variables w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . . 7 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
2	1	cnfldtopon 22586	. . . . . 6 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
3	2	a1i 11	. . . . 5 |- ( ph -> ( TopOpen ` ℂfld ) e. (TopOn ` CC ) )
4	3	cnmptid 21464	. . . . 5 |- ( ph -> ( x e. CC |-> x ) e. ( ( TopOpen ` ℂfld ) Cn ( TopOpen ` ℂfld ) ) )
5		rmulccn.2	. . . . . . 7 |- ( ph -> C e. RR )
6	5	recnd 10068	. . . . . 6 |- ( ph -> C e. CC )
7	3, 3, 6	cnmptc 21465	. . . . 5 |- ( ph -> ( x e. CC |-> C ) e. ( ( TopOpen ` ℂfld ) Cn ( TopOpen ` ℂfld ) ) )
8		ax-mulf 10016	. . . . . . . . 9 |- x. : ( CC X. CC ) --> CC
9		ffn 6045	. . . . . . . . 9 |- ( x. : ( CC X. CC ) --> CC -> x. Fn ( CC X. CC ) )
10	8, 9	ax-mp 5	. . . . . . . 8 |- x. Fn ( CC X. CC )
11		fnov 6768	. . . . . . . 8 |- ( x. Fn ( CC X. CC ) <-> x. = ( y e. CC , z e. CC |-> ( y x. z ) ) )
12	10, 11	mpbi 220	. . . . . . 7 |- x. = ( y e. CC , z e. CC |-> ( y x. z ) )
13	1	mulcn 22670	. . . . . . 7 |- x. e. ( ( ( TopOpen ` ℂfld ) tX ( TopOpen ` ℂfld ) ) Cn ( TopOpen ` ℂfld ) )
14	12, 13	eqeltrri 2698	. . . . . 6 |- ( y e. CC , z e. CC |-> ( y x. z ) ) e. ( ( ( TopOpen ` ℂfld ) tX ( TopOpen ` ℂfld ) ) Cn ( TopOpen ` ℂfld ) )
15	14	a1i 11	. . . . 5 |- ( ph -> ( y e. CC , z e. CC |-> ( y x. z ) ) e. ( ( ( TopOpen ` ℂfld ) tX ( TopOpen ` ℂfld ) ) Cn ( TopOpen ` ℂfld ) ) )
16		oveq12 6659	. . . . 5 |- ( ( y = x /\ z = C ) -> ( y x. z ) = ( x x. C ) )
17	3, 4, 7, 3, 3, 15, 16	cnmpt12 21470	. . . 4 |- ( ph -> ( x e. CC |-> ( x x. C ) ) e. ( ( TopOpen ` ℂfld ) Cn ( TopOpen ` ℂfld ) ) )
18		ax-resscn 9993	. . . 4 |- RR C_ CC
19	2	toponunii 20721	. . . . 5 |- CC = U. ( TopOpen ` ℂfld )
20	19	cnrest 21089	. . . 4 |- ( ( ( x e. CC |-> ( x x. C ) ) e. ( ( TopOpen ` ℂfld ) Cn ( TopOpen ` ℂfld ) ) /\ RR C_ CC ) -> ( ( x e. CC |-> ( x x. C ) ) |` RR ) e. ( ( ( TopOpen ` ℂfld ) ↾t RR ) Cn ( TopOpen ` ℂfld ) ) )
21	17, 18, 20	sylancl 694	. . 3 |- ( ph -> ( ( x e. CC |-> ( x x. C ) ) |` RR ) e. ( ( ( TopOpen ` ℂfld ) ↾t RR ) Cn ( TopOpen ` ℂfld ) ) )
22		simpr 477	. . . . . . . . 9 |- ( ( ph /\ x e. CC ) -> x e. CC )
23	6	adantr 481	. . . . . . . . 9 |- ( ( ph /\ x e. CC ) -> C e. CC )
24	22, 23	mulcld 10060	. . . . . . . 8 |- ( ( ph /\ x e. CC ) -> ( x x. C ) e. CC )
25	24	ralrimiva 2966	. . . . . . 7 |- ( ph -> A. x e. CC ( x x. C ) e. CC )
26		eqid 2622	. . . . . . . 8 |- ( x e. CC |-> ( x x. C ) ) = ( x e. CC |-> ( x x. C ) )
27	26	fnmpt 6020	. . . . . . 7 |- ( A. x e. CC ( x x. C ) e. CC -> ( x e. CC |-> ( x x. C ) ) Fn CC )
28	25, 27	syl 17	. . . . . 6 |- ( ph -> ( x e. CC |-> ( x x. C ) ) Fn CC )
29		fnssres 6004	. . . . . 6 |- ( ( ( x e. CC |-> ( x x. C ) ) Fn CC /\ RR C_ CC ) -> ( ( x e. CC |-> ( x x. C ) ) |` RR ) Fn RR )
30	28, 18, 29	sylancl 694	. . . . 5 |- ( ph -> ( ( x e. CC |-> ( x x. C ) ) |` RR ) Fn RR )
31		simpr 477	. . . . . . . 8 |- ( ( ph /\ w e. RR ) -> w e. RR )
32		fvres 6207	. . . . . . . . 9 |- ( w e. RR -> ( ( ( x e. CC |-> ( x x. C ) ) |` RR ) ` w ) = ( ( x e. CC |-> ( x x. C ) ) ` w ) )
33		recn 10026	. . . . . . . . . 10 |- ( w e. RR -> w e. CC )
34		oveq1 6657	. . . . . . . . . . 11 |- ( x = w -> ( x x. C ) = ( w x. C ) )
35		ovex 6678	. . . . . . . . . . 11 |- ( w x. C ) e. _V
36	34, 26, 35	fvmpt 6282	. . . . . . . . . 10 |- ( w e. CC -> ( ( x e. CC |-> ( x x. C ) ) ` w ) = ( w x. C ) )
37	33, 36	syl 17	. . . . . . . . 9 |- ( w e. RR -> ( ( x e. CC |-> ( x x. C ) ) ` w ) = ( w x. C ) )
38	32, 37	eqtrd 2656	. . . . . . . 8 |- ( w e. RR -> ( ( ( x e. CC |-> ( x x. C ) ) |` RR ) ` w ) = ( w x. C ) )
39	31, 38	syl 17	. . . . . . 7 |- ( ( ph /\ w e. RR ) -> ( ( ( x e. CC |-> ( x x. C ) ) |` RR ) ` w ) = ( w x. C ) )
40	5	adantr 481	. . . . . . . 8 |- ( ( ph /\ w e. RR ) -> C e. RR )
41	31, 40	remulcld 10070	. . . . . . 7 |- ( ( ph /\ w e. RR ) -> ( w x. C ) e. RR )
42	39, 41	eqeltrd 2701	. . . . . 6 |- ( ( ph /\ w e. RR ) -> ( ( ( x e. CC |-> ( x x. C ) ) |` RR ) ` w ) e. RR )
43	42	ralrimiva 2966	. . . . 5 |- ( ph -> A. w e. RR ( ( ( x e. CC |-> ( x x. C ) ) |` RR ) ` w ) e. RR )
44		fnfvrnss 6390	. . . . 5 |- ( ( ( ( x e. CC |-> ( x x. C ) ) |` RR ) Fn RR /\ A. w e. RR ( ( ( x e. CC |-> ( x x. C ) ) |` RR ) ` w ) e. RR ) -> ran ( ( x e. CC |-> ( x x. C ) ) |` RR ) C_ RR )
45	30, 43, 44	syl2anc 693	. . . 4 |- ( ph -> ran ( ( x e. CC |-> ( x x. C ) ) |` RR ) C_ RR )
46	18	a1i 11	. . . 4 |- ( ph -> RR C_ CC )
47		cnrest2 21090	. . . 4 |- ( ( ( TopOpen ` ℂfld ) e. (TopOn ` CC ) /\ ran ( ( x e. CC |-> ( x x. C ) ) |` RR ) C_ RR /\ RR C_ CC ) -> ( ( ( x e. CC |-> ( x x. C ) ) |` RR ) e. ( ( ( TopOpen ` ℂfld ) ↾t RR ) Cn ( TopOpen ` ℂfld ) ) <-> ( ( x e. CC |-> ( x x. C ) ) |` RR ) e. ( ( ( TopOpen ` ℂfld ) ↾t RR ) Cn ( ( TopOpen ` ℂfld ) ↾t RR ) ) ) )
48	3, 45, 46, 47	syl3anc 1326	. . 3 |- ( ph -> ( ( ( x e. CC |-> ( x x. C ) ) |` RR ) e. ( ( ( TopOpen ` ℂfld ) ↾t RR ) Cn ( TopOpen ` ℂfld ) ) <-> ( ( x e. CC |-> ( x x. C ) ) |` RR ) e. ( ( ( TopOpen ` ℂfld ) ↾t RR ) Cn ( ( TopOpen ` ℂfld ) ↾t RR ) ) ) )
49	21, 48	mpbid 222	. 2 |- ( ph -> ( ( x e. CC |-> ( x x. C ) ) |` RR ) e. ( ( ( TopOpen ` ℂfld ) ↾t RR ) Cn ( ( TopOpen ` ℂfld ) ↾t RR ) ) )
50		resmpt 5449	. . 3 |- ( RR C_ CC -> ( ( x e. CC |-> ( x x. C ) ) |` RR ) = ( x e. RR |-> ( x x. C ) ) )
51	18, 50	ax-mp 5	. 2 |- ( ( x e. CC |-> ( x x. C ) ) |` RR ) = ( x e. RR |-> ( x x. C ) )
52		rmulccn.1	. . . . 5 |- J = ( topGen ` ran (,) )
53	1	tgioo2 22606	. . . . 5 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
54	52, 53	eqtri 2644	. . . 4 |- J = ( ( TopOpen ` ℂfld ) ↾t RR )
55	54, 54	oveq12i 6662	. . 3 |- ( J Cn J ) = ( ( ( TopOpen ` ℂfld ) ↾t RR ) Cn ( ( TopOpen ` ℂfld ) ↾t RR ) )
56	55	eqcomi 2631	. 2 |- ( ( ( TopOpen ` ℂfld ) ↾t RR ) Cn ( ( TopOpen ` ℂfld ) ↾t RR ) ) = ( J Cn J )
57	49, 51, 56	3eltr3g 2717	1 |- ( ph -> ( x e. RR |-> ( x x. C ) ) e. ( J Cn J ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   |-> cmpt 4729   X. cxp 5112   ran crn 5115   |` cres 5116   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   CCcc 9934   RRcr 9935   x. cmul 9941   (,)cioo 12175   ↾_t crest 16081   TopOpenctopn 16082   topGenctg 16098  ℂ_fldccnfld 19746  TopOnctopon 20715   Cn ccn 21028   tX ctx 21363

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  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  ax-mulf 10016

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-int 4476  df-iun 4522  df-iin 4523  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  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-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cn 21031  df-cnp 21032  df-tx 21365  df-hmeo 21558  df-xms 22125  df-ms 22126  df-tms 22127

This theorem is referenced by:  rrvmulc  30515