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Theorem mulerpqlem 9350
Description: Lemma for mulerpq 9352. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
mulerpqlem  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( A  ~Q  B  <->  ( A  .pQ  C )  ~Q  ( B 
.pQ  C ) ) )

Proof of Theorem mulerpqlem
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xp1st 6829 . . . . 5  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
213ad2ant1 1017 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 1st `  A )  e.  N. )
3 xp1st 6829 . . . . 5  |-  ( C  e.  ( N.  X.  N. )  ->  ( 1st `  C )  e.  N. )
433ad2ant3 1019 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 1st `  C )  e.  N. )
5 mulclpi 9288 . . . 4  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 1st `  C )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 1st `  C ) )  e. 
N. )
62, 4, 5syl2anc 661 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( ( 1st `  A )  .N  ( 1st `  C
) )  e.  N. )
7 xp2nd 6830 . . . . 5  |-  ( A  e.  ( N.  X.  N. )  ->  ( 2nd `  A )  e.  N. )
873ad2ant1 1017 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 2nd `  A )  e.  N. )
9 xp2nd 6830 . . . . 5  |-  ( C  e.  ( N.  X.  N. )  ->  ( 2nd `  C )  e.  N. )
1093ad2ant3 1019 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 2nd `  C )  e.  N. )
11 mulclpi 9288 . . . 4  |-  ( ( ( 2nd `  A
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  A
)  .N  ( 2nd `  C ) )  e. 
N. )
128, 10, 11syl2anc 661 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( ( 2nd `  A )  .N  ( 2nd `  C
) )  e.  N. )
13 xp1st 6829 . . . . 5  |-  ( B  e.  ( N.  X.  N. )  ->  ( 1st `  B )  e.  N. )
14133ad2ant2 1018 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 1st `  B )  e.  N. )
15 mulclpi 9288 . . . 4  |-  ( ( ( 1st `  B
)  e.  N.  /\  ( 1st `  C )  e.  N. )  -> 
( ( 1st `  B
)  .N  ( 1st `  C ) )  e. 
N. )
1614, 4, 15syl2anc 661 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( ( 1st `  B )  .N  ( 1st `  C
) )  e.  N. )
17 xp2nd 6830 . . . . 5  |-  ( B  e.  ( N.  X.  N. )  ->  ( 2nd `  B )  e.  N. )
18173ad2ant2 1018 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 2nd `  B )  e.  N. )
19 mulclpi 9288 . . . 4  |-  ( ( ( 2nd `  B
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
2018, 10, 19syl2anc 661 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( ( 2nd `  B )  .N  ( 2nd `  C
) )  e.  N. )
21 enqbreq 9314 . . 3  |-  ( ( ( ( ( 1st `  A )  .N  ( 1st `  C ) )  e.  N.  /\  (
( 2nd `  A
)  .N  ( 2nd `  C ) )  e. 
N. )  /\  (
( ( 1st `  B
)  .N  ( 1st `  C ) )  e. 
N.  /\  ( ( 2nd `  B )  .N  ( 2nd `  C
) )  e.  N. ) )  ->  ( <. ( ( 1st `  A
)  .N  ( 1st `  C ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  C ) ) >.  ~Q  <. ( ( 1st `  B )  .N  ( 1st `  C ) ) ,  ( ( 2nd `  B )  .N  ( 2nd `  C ) )
>. 
<->  ( ( ( 1st `  A )  .N  ( 1st `  C ) )  .N  ( ( 2nd `  B )  .N  ( 2nd `  C ) ) )  =  ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  B
)  .N  ( 1st `  C ) ) ) ) )
226, 12, 16, 20, 21syl22anc 1229 . 2  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( <. ( ( 1st `  A
)  .N  ( 1st `  C ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  C ) ) >.  ~Q  <. ( ( 1st `  B )  .N  ( 1st `  C ) ) ,  ( ( 2nd `  B )  .N  ( 2nd `  C ) )
>. 
<->  ( ( ( 1st `  A )  .N  ( 1st `  C ) )  .N  ( ( 2nd `  B )  .N  ( 2nd `  C ) ) )  =  ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  B
)  .N  ( 1st `  C ) ) ) ) )
23 mulpipq2 9334 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  C  e.  ( N.  X.  N. ) )  ->  ( A  .pQ  C )  = 
<. ( ( 1st `  A
)  .N  ( 1st `  C ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  C ) ) >.
)
24233adant2 1015 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( A  .pQ  C )  =  <. ( ( 1st `  A
)  .N  ( 1st `  C ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  C ) ) >.
)
25 mulpipq2 9334 . . . 4  |-  ( ( B  e.  ( N. 
X.  N. )  /\  C  e.  ( N.  X.  N. ) )  ->  ( B  .pQ  C )  = 
<. ( ( 1st `  B
)  .N  ( 1st `  C ) ) ,  ( ( 2nd `  B
)  .N  ( 2nd `  C ) ) >.
)
26253adant1 1014 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( B  .pQ  C )  =  <. ( ( 1st `  B
)  .N  ( 1st `  C ) ) ,  ( ( 2nd `  B
)  .N  ( 2nd `  C ) ) >.
)
2724, 26breq12d 4469 . 2  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( ( A  .pQ  C )  ~Q  ( B  .pQ  C )  <->  <. ( ( 1st `  A
)  .N  ( 1st `  C ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  C ) ) >.  ~Q  <. ( ( 1st `  B )  .N  ( 1st `  C ) ) ,  ( ( 2nd `  B )  .N  ( 2nd `  C ) )
>. ) )
28 enqbreq2 9315 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  ~Q  B  <->  ( ( 1st `  A )  .N  ( 2nd `  B
) )  =  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) ) )
29283adant3 1016 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( A  ~Q  B  <->  ( ( 1st `  A )  .N  ( 2nd `  B ) )  =  ( ( 1st `  B )  .N  ( 2nd `  A ) ) ) )
30 mulclpi 9288 . . . . 5  |-  ( ( ( 1st `  C
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 1st `  C
)  .N  ( 2nd `  C ) )  e. 
N. )
314, 10, 30syl2anc 661 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( ( 1st `  C )  .N  ( 2nd `  C
) )  e.  N. )
32 mulclpi 9288 . . . . 5  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
332, 18, 32syl2anc 661 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( ( 1st `  A )  .N  ( 2nd `  B
) )  e.  N. )
34 mulcanpi 9295 . . . 4  |-  ( ( ( ( 1st `  C
)  .N  ( 2nd `  C ) )  e. 
N.  /\  ( ( 1st `  A )  .N  ( 2nd `  B
) )  e.  N. )  ->  ( ( ( ( 1st `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 1st `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) )  <-> 
( ( 1st `  A
)  .N  ( 2nd `  B ) )  =  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) ) )
3531, 33, 34syl2anc 661 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( (
( ( 1st `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 1st `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) )  <-> 
( ( 1st `  A
)  .N  ( 2nd `  B ) )  =  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) ) )
36 mulcompi 9291 . . . . . 6  |-  ( ( ( 1st `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 1st `  A )  .N  ( 2nd `  B
) )  .N  (
( 1st `  C
)  .N  ( 2nd `  C ) ) )
37 fvex 5882 . . . . . . 7  |-  ( 1st `  A )  e.  _V
38 fvex 5882 . . . . . . 7  |-  ( 2nd `  B )  e.  _V
39 fvex 5882 . . . . . . 7  |-  ( 1st `  C )  e.  _V
40 mulcompi 9291 . . . . . . 7  |-  ( x  .N  y )  =  ( y  .N  x
)
41 mulasspi 9292 . . . . . . 7  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
42 fvex 5882 . . . . . . 7  |-  ( 2nd `  C )  e.  _V
4337, 38, 39, 40, 41, 42caov4 6505 . . . . . 6  |-  ( ( ( 1st `  A
)  .N  ( 2nd `  B ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  C ) ) )  =  ( ( ( 1st `  A )  .N  ( 1st `  C
) )  .N  (
( 2nd `  B
)  .N  ( 2nd `  C ) ) )
4436, 43eqtri 2486 . . . . 5  |-  ( ( ( 1st `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 1st `  A )  .N  ( 1st `  C
) )  .N  (
( 2nd `  B
)  .N  ( 2nd `  C ) ) )
45 mulcompi 9291 . . . . . 6  |-  ( ( ( 1st `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) )  =  ( ( ( 1st `  B )  .N  ( 2nd `  A
) )  .N  (
( 1st `  C
)  .N  ( 2nd `  C ) ) )
46 fvex 5882 . . . . . . 7  |-  ( 1st `  B )  e.  _V
47 fvex 5882 . . . . . . 7  |-  ( 2nd `  A )  e.  _V
4846, 47, 39, 40, 41, 42caov4 6505 . . . . . 6  |-  ( ( ( 1st `  B
)  .N  ( 2nd `  A ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  C ) ) )  =  ( ( ( 1st `  B )  .N  ( 1st `  C
) )  .N  (
( 2nd `  A
)  .N  ( 2nd `  C ) ) )
49 mulcompi 9291 . . . . . 6  |-  ( ( ( 1st `  B
)  .N  ( 1st `  C ) )  .N  ( ( 2nd `  A
)  .N  ( 2nd `  C ) ) )  =  ( ( ( 2nd `  A )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 1st `  C ) ) )
5045, 48, 493eqtri 2490 . . . . 5  |-  ( ( ( 1st `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) )  =  ( ( ( 2nd `  A )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 1st `  C ) ) )
5144, 50eqeq12i 2477 . . . 4  |-  ( ( ( ( 1st `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 1st `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) )  <-> 
( ( ( 1st `  A )  .N  ( 1st `  C ) )  .N  ( ( 2nd `  B )  .N  ( 2nd `  C ) ) )  =  ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  B
)  .N  ( 1st `  C ) ) ) )
5251a1i 11 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( (
( ( 1st `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 1st `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) )  <-> 
( ( ( 1st `  A )  .N  ( 1st `  C ) )  .N  ( ( 2nd `  B )  .N  ( 2nd `  C ) ) )  =  ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  B
)  .N  ( 1st `  C ) ) ) ) )
5329, 35, 523bitr2d 281 . 2  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( A  ~Q  B  <->  ( ( ( 1st `  A )  .N  ( 1st `  C
) )  .N  (
( 2nd `  B
)  .N  ( 2nd `  C ) ) )  =  ( ( ( 2nd `  A )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 1st `  C ) ) ) ) )
5422, 27, 533bitr4rd 286 1  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( A  ~Q  B  <->  ( A  .pQ  C )  ~Q  ( B 
.pQ  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 973    = wceq 1395    e. wcel 1819   <.cop 4038   class class class wbr 4456    X. cxp 5006   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798   N.cnpi 9239    .N cmi 9241    .pQ cmpq 9244    ~Q ceq 9246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-oadd 7152  df-omul 7153  df-ni 9267  df-mi 9269  df-mpq 9304  df-enq 9306
This theorem is referenced by:  mulerpq  9352
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