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Theorem srgfcl 12969
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.
StepHypRef 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 )
42, 3srgcl 12966 . . . . . . 7  |-  ( ( R  e. SRing  /\  a  e.  B  /\  b  e.  B )  ->  (
a  .x.  b )  e.  B )
543expb 1204 . . . . . 6  |-  ( ( R  e. SRing  /\  (
a  e.  B  /\  b  e.  B )
)  ->  ( a  .x.  b )  e.  B
)
65ralrimivva 2559 . . . . 5  |-  ( R  e. SRing  ->  A. a  e.  B  A. b  e.  B  ( a  .x.  b
)  e.  B )
7 fveq2 5510 . . . . . . . 8  |-  ( c  =  <. a ,  b
>.  ->  (  .x.  `  c
)  =  (  .x.  ` 
<. a ,  b >.
) )
87eleq1d 2246 . . . . . . 7  |-  ( c  =  <. a ,  b
>.  ->  ( (  .x.  `  c )  e.  B  <->  ( 
.x.  `  <. a ,  b >. )  e.  B
) )
9 df-ov 5871 . . . . . . . . 9  |-  ( a 
.x.  b )  =  (  .x.  `  <. a ,  b >. )
109eqcomi 2181 . . . . . . . 8  |-  (  .x.  ` 
<. a ,  b >.
)  =  ( a 
.x.  b )
1110eleq1i 2243 . . . . . . 7  |-  ( ( 
.x.  `  <. a ,  b >. )  e.  B  <->  ( a  .x.  b )  e.  B )
128, 11bitrdi 196 . . . . . 6  |-  ( c  =  <. a ,  b
>.  ->  ( (  .x.  `  c )  e.  B  <->  ( a  .x.  b )  e.  B ) )
1312ralxp 4765 . . . . 5  |-  ( A. c  e.  ( B  X.  B ) (  .x.  `  c )  e.  B  <->  A. a  e.  B  A. b  e.  B  (
a  .x.  b )  e.  B )
146, 13sylibr 134 . . . 4  |-  ( R  e. SRing  ->  A. c  e.  ( B  X.  B ) (  .x.  `  c
)  e.  B )
1514adantr 276 . . 3  |-  ( ( R  e. SRing  /\  .x.  Fn  ( B  X.  B
) )  ->  A. c  e.  ( B  X.  B
) (  .x.  `  c
)  e.  B )
16 fnfvrnss 5671 . . 3  |-  ( ( 
.x.  Fn  ( B  X.  B )  /\  A. c  e.  ( B  X.  B ) (  .x.  `  c )  e.  B
)  ->  ran  .x.  C_  B
)
171, 15, 16syl2anc 411 . 2  |-  ( ( R  e. SRing  /\  .x.  Fn  ( B  X.  B
) )  ->  ran  .x.  C_  B )
18 df-f 5215 . 2  |-  (  .x.  : ( B  X.  B
) --> B  <->  (  .x.  Fn  ( B  X.  B
)  /\  ran  .x.  C_  B
) )
191, 17, 18sylanbrc 417 1  |-  ( ( R  e. SRing  /\  .x.  Fn  ( B  X.  B
) )  ->  .x.  :
( B  X.  B
) --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   A.wral 2455    C_ wss 3129   <.cop 3594    X. cxp 4620   ran crn 4623    Fn wfn 5206   -->wf 5207   ` cfv 5211  (class class class)co 5868   Basecbs 12432   .rcmulr 12506  SRingcsrg 12959
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-addcom 7889  ax-addass 7891  ax-i2m1 7894  ax-0lt1 7895  ax-0id 7897  ax-rnegex 7898  ax-pre-ltirr 7901  ax-pre-ltadd 7905
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-pnf 7971  df-mnf 7972  df-ltxr 7974  df-inn 8896  df-2 8954  df-3 8955  df-ndx 12435  df-slot 12436  df-base 12438  df-sets 12439  df-plusg 12518  df-mulr 12519  df-0g 12642  df-mgm 12654  df-sgrp 12687  df-mnd 12697  df-mgp 12945  df-srg 12960
This theorem is referenced by: (None)
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