Theorem srgfcl 14134

Description: Functionality of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by AV, 24-Aug-2021.)

Hypotheses

Ref	Expression
srgfcl.b	|- B = ( Base ` R )
srgfcl.t	|- .x. = ( .r ` R )

Assertion

Ref	Expression
srgfcl	|- ( ( R e. SRing /\ .x. Fn ( B X. B ) ) -> .x. : ( B X. B ) --> B )

Proof of Theorem srgfcl

Dummy variables a b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. 2 |- ( ( R e. SRing /\ .x. Fn ( B X. B ) ) -> .x. Fn ( B X. B ) )
2		srgfcl.b	. . . . . . . 8 |- B = ( Base ` R )
3		srgfcl.t	. . . . . . . 8 |- .x. = ( .r ` R )
4	2, 3	srgcl 14131	. . . . . . 7 |- ( ( R e. SRing /\ a e. B /\ b e. B ) -> ( a .x. b ) e. B )
5	4	3expb 1231	. . . . . 6 |- ( ( R e. SRing /\ ( a e. B /\ b e. B ) ) -> ( a .x. b ) e. B )
6	5	ralrimivva 2626	. . . . 5 |- ( R e. SRing -> A. a e. B A. b e. B ( a .x. b ) e. B )
7		fveq2 5672	. . . . . . . 8 |- ( c = <. a , b >. -> ( .x. ` c ) = ( .x. ` <. a , b >. ) )
8	7	eleq1d 2303	. . . . . . 7 |- ( c = <. a , b >. -> ( ( .x. ` c ) e. B <-> ( .x. ` <. a , b >. ) e. B ) )
9		df-ov 6055	. . . . . . . . 9 |- ( a .x. b ) = ( .x. ` <. a , b >. )
10	9	eqcomi 2238	. . . . . . . 8 |- ( .x. ` <. a , b >. ) = ( a .x. b )
11	10	eleq1i 2300	. . . . . . 7 |- ( ( .x. ` <. a , b >. ) e. B <-> ( a .x. b ) e. B )
12	8, 11	bitrdi 196	. . . . . 6 |- ( c = <. a , b >. -> ( ( .x. ` c ) e. B <-> ( a .x. b ) e. B ) )
13	12	ralxp 4900	. . . . 5 |- ( A. c e. ( B X. B ) ( .x. ` c ) e. B <-> A. a e. B A. b e. B ( a .x. b ) e. B )
14	6, 13	sylibr 134	. . . 4 |- ( R e. SRing -> A. c e. ( B X. B ) ( .x. ` c ) e. B )
15	14	adantr 276	. . 3 |- ( ( R e. SRing /\ .x. Fn ( B X. B ) ) -> A. c e. ( B X. B ) ( .x. ` c ) e. B )
16		fnfvrnss 5839	. . 3 |- ( ( .x. Fn ( B X. B ) /\ A. c e. ( B X. B ) ( .x. ` c ) e. B ) -> ran .x. C_ B )
17	1, 15, 16	syl2anc 411	. 2 |- ( ( R e. SRing /\ .x. Fn ( B X. B ) ) -> ran .x. C_ B )
18		df-f 5358	. 2 |- ( .x. : ( B X. B ) --> B <-> ( .x. Fn ( B X. B ) /\ ran .x. C_ B ) )
19	1, 17, 18	sylanbrc 417	1 |- ( ( R e. SRing /\ .x. Fn ( B X. B ) ) -> .x. : ( B X. B ) --> B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   A.wral 2522   C_ wss 3213   <.cop 3694   X. cxp 4749   ran crn 4752   Fn wfn 5349   -->wf 5350   ` cfv 5354  (class class class)co 6052   Basecbs 13229   .rcmulr 13308  SRingcsrg 14124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-pre-ltirr 8241  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-pnf 8312  df-mnf 8313  df-ltxr 8315  df-inn 9240  df-2 9298  df-3 9299  df-ndx 13232  df-slot 13233  df-base 13235  df-sets 13236  df-plusg 13320  df-mulr 13321  df-0g 13488  df-mgm 13586  df-sgrp 13632  df-mnd 13647  df-mgp 14082  df-srg 14125

This theorem is referenced by: (None)