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Theorem remulext1 8066
Description: Left extensionality for multiplication. (Contributed by Jim Kingdon, 19-Feb-2020.)
Assertion
Ref Expression
remulext1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  x.  C
) #  ( B  x.  C )  ->  A #  B ) )

Proof of Theorem remulext1
StepHypRef Expression
1 simp1 943 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A  e.  RR )
2 simp3 945 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  C  e.  RR )
31, 2remulcld 7508 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  x.  C )  e.  RR )
4 simp2 944 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  B  e.  RR )
54, 2remulcld 7508 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  x.  C )  e.  RR )
6 reaplt 8055 . . 3  |-  ( ( ( A  x.  C
)  e.  RR  /\  ( B  x.  C
)  e.  RR )  ->  ( ( A  x.  C ) #  ( B  x.  C )  <-> 
( ( A  x.  C )  <  ( B  x.  C )  \/  ( B  x.  C
)  <  ( A  x.  C ) ) ) )
73, 5, 6syl2anc 403 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  x.  C
) #  ( B  x.  C )  <->  ( ( A  x.  C )  <  ( B  x.  C
)  \/  ( B  x.  C )  < 
( A  x.  C
) ) ) )
8 ax-pre-mulext 7453 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  x.  C
)  <RR  ( B  x.  C )  ->  ( A  <RR  B  \/  B  <RR  A ) ) )
9 ltxrlt 7542 . . . . 5  |-  ( ( ( A  x.  C
)  e.  RR  /\  ( B  x.  C
)  e.  RR )  ->  ( ( A  x.  C )  < 
( B  x.  C
)  <->  ( A  x.  C )  <RR  ( B  x.  C ) ) )
103, 5, 9syl2anc 403 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  x.  C
)  <  ( B  x.  C )  <->  ( A  x.  C )  <RR  ( B  x.  C ) ) )
11 reaplt 8055 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  <->  ( A  <  B  \/  B  < 
A ) ) )
121, 4, 11syl2anc 403 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A #  B  <->  ( A  < 
B  \/  B  < 
A ) ) )
13 ltxrlt 7542 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  A 
<RR  B ) )
141, 4, 13syl2anc 403 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  A  <RR  B ) )
15 ltxrlt 7542 . . . . . . 7  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B  <  A  <->  B 
<RR  A ) )
164, 1, 15syl2anc 403 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  <  A  <->  B  <RR  A ) )
1714, 16orbi12d 742 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <  B  \/  B  <  A )  <-> 
( A  <RR  B  \/  B  <RR  A ) ) )
1812, 17bitrd 186 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A #  B  <->  ( A  <RR  B  \/  B  <RR  A ) ) )
198, 10, 183imtr4d 201 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  x.  C
)  <  ( B  x.  C )  ->  A #  B ) )
20 ax-pre-mulext 7453 . . . . 5  |-  ( ( B  e.  RR  /\  A  e.  RR  /\  C  e.  RR )  ->  (
( B  x.  C
)  <RR  ( A  x.  C )  ->  ( B  <RR  A  \/  A  <RR  B ) ) )
21203com12 1147 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( B  x.  C
)  <RR  ( A  x.  C )  ->  ( B  <RR  A  \/  A  <RR  B ) ) )
22 ltxrlt 7542 . . . . 5  |-  ( ( ( B  x.  C
)  e.  RR  /\  ( A  x.  C
)  e.  RR )  ->  ( ( B  x.  C )  < 
( A  x.  C
)  <->  ( B  x.  C )  <RR  ( A  x.  C ) ) )
235, 3, 22syl2anc 403 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( B  x.  C
)  <  ( A  x.  C )  <->  ( B  x.  C )  <RR  ( A  x.  C ) ) )
24 orcom 682 . . . . 5  |-  ( ( A  <RR  B  \/  B  <RR  A )  <->  ( B  <RR  A  \/  A  <RR  B ) )
2518, 24syl6bb 194 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A #  B  <->  ( B  <RR  A  \/  A  <RR  B ) ) )
2621, 23, 253imtr4d 201 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( B  x.  C
)  <  ( A  x.  C )  ->  A #  B ) )
2719, 26jaod 672 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( ( A  x.  C )  <  ( B  x.  C )  \/  ( B  x.  C
)  <  ( A  x.  C ) )  ->  A #  B ) )
287, 27sylbid 148 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  x.  C
) #  ( B  x.  C )  ->  A #  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    \/ wo 664    /\ w3a 924    e. wcel 1438   class class class wbr 3843  (class class class)co 5644   RRcr 7339    <RR cltrr 7344    x. cmul 7345    < clt 7512   # cap 8048
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034  ax-un 4258  ax-setind 4351  ax-cnex 7426  ax-resscn 7427  ax-1cn 7428  ax-1re 7429  ax-icn 7430  ax-addcl 7431  ax-addrcl 7432  ax-mulcl 7433  ax-mulrcl 7434  ax-addcom 7435  ax-mulcom 7436  ax-addass 7437  ax-mulass 7438  ax-distr 7439  ax-i2m1 7440  ax-0lt1 7441  ax-1rid 7442  ax-0id 7443  ax-rnegex 7444  ax-precex 7445  ax-cnre 7446  ax-pre-ltirr 7447  ax-pre-lttrn 7449  ax-pre-apti 7450  ax-pre-ltadd 7451  ax-pre-mulgt0 7452  ax-pre-mulext 7453
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-br 3844  df-opab 3898  df-id 4118  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-iota 4975  df-fun 5012  df-fv 5018  df-riota 5600  df-ov 5647  df-oprab 5648  df-mpt2 5649  df-pnf 7514  df-mnf 7515  df-ltxr 7517  df-sub 7645  df-neg 7646  df-reap 8042  df-ap 8049
This theorem is referenced by:  remulext2  8067  mulext1  8079
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