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Theorem rhmdvdsr 28593
Description: A ring homomorphism preserves the divisibility relation. (Contributed by Thierry Arnoux, 22-Oct-2017.)
Hypotheses
Ref Expression
rhmdvdsr.x  |-  X  =  ( Base `  R
)
rhmdvdsr.m  |-  .||  =  (
||r `  R )
rhmdvdsr.n  |-  ./  =  ( ||r `
 S )
Assertion
Ref Expression
rhmdvdsr  |-  ( ( ( F  e.  ( R RingHom  S )  /\  A  e.  X  /\  B  e.  X )  /\  A  .|| 
B )  ->  ( F `  A )  ./  ( F `  B
) )

Proof of Theorem rhmdvdsr
Dummy variables  y 
c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl1 1012 . . 3  |-  ( ( ( F  e.  ( R RingHom  S )  /\  A  e.  X  /\  B  e.  X )  /\  A  .|| 
B )  ->  F  e.  ( R RingHom  S )
)
2 simpl2 1013 . . 3  |-  ( ( ( F  e.  ( R RingHom  S )  /\  A  e.  X  /\  B  e.  X )  /\  A  .|| 
B )  ->  A  e.  X )
3 rhmdvdsr.x . . . . 5  |-  X  =  ( Base `  R
)
4 eqid 2453 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
53, 4rhmf 17966 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  F : X
--> ( Base `  S
) )
65ffvelrnda 6027 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  X )  ->  ( F `  A )  e.  ( Base `  S
) )
71, 2, 6syl2anc 667 . 2  |-  ( ( ( F  e.  ( R RingHom  S )  /\  A  e.  X  /\  B  e.  X )  /\  A  .|| 
B )  ->  ( F `  A )  e.  ( Base `  S
) )
8 simpll1 1048 . . . . . 6  |-  ( ( ( ( F  e.  ( R RingHom  S )  /\  A  e.  X  /\  B  e.  X
)  /\  A  .||  B )  /\  c  e.  X
)  ->  F  e.  ( R RingHom  S ) )
9 simpr 463 . . . . . 6  |-  ( ( ( ( F  e.  ( R RingHom  S )  /\  A  e.  X  /\  B  e.  X
)  /\  A  .||  B )  /\  c  e.  X
)  ->  c  e.  X )
105ffvelrnda 6027 . . . . . 6  |-  ( ( F  e.  ( R RingHom  S )  /\  c  e.  X )  ->  ( F `  c )  e.  ( Base `  S
) )
118, 9, 10syl2anc 667 . . . . 5  |-  ( ( ( ( F  e.  ( R RingHom  S )  /\  A  e.  X  /\  B  e.  X
)  /\  A  .||  B )  /\  c  e.  X
)  ->  ( F `  c )  e.  (
Base `  S )
)
1211ralrimiva 2804 . . . 4  |-  ( ( ( F  e.  ( R RingHom  S )  /\  A  e.  X  /\  B  e.  X )  /\  A  .|| 
B )  ->  A. c  e.  X  ( F `  c )  e.  (
Base `  S )
)
132adantr 467 . . . . . . 7  |-  ( ( ( ( F  e.  ( R RingHom  S )  /\  A  e.  X  /\  B  e.  X
)  /\  A  .||  B )  /\  c  e.  X
)  ->  A  e.  X )
14 eqid 2453 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
15 eqid 2453 . . . . . . . 8  |-  ( .r
`  S )  =  ( .r `  S
)
163, 14, 15rhmmul 17967 . . . . . . 7  |-  ( ( F  e.  ( R RingHom  S )  /\  c  e.  X  /\  A  e.  X )  ->  ( F `  ( c
( .r `  R
) A ) )  =  ( ( F `
 c ) ( .r `  S ) ( F `  A
) ) )
178, 9, 13, 16syl3anc 1269 . . . . . 6  |-  ( ( ( ( F  e.  ( R RingHom  S )  /\  A  e.  X  /\  B  e.  X
)  /\  A  .||  B )  /\  c  e.  X
)  ->  ( F `  ( c ( .r
`  R ) A ) )  =  ( ( F `  c
) ( .r `  S ) ( F `
 A ) ) )
1817ralrimiva 2804 . . . . 5  |-  ( ( ( F  e.  ( R RingHom  S )  /\  A  e.  X  /\  B  e.  X )  /\  A  .|| 
B )  ->  A. c  e.  X  ( F `  ( c ( .r
`  R ) A ) )  =  ( ( F `  c
) ( .r `  S ) ( F `
 A ) ) )
19 simpr 463 . . . . . 6  |-  ( ( ( F  e.  ( R RingHom  S )  /\  A  e.  X  /\  B  e.  X )  /\  A  .|| 
B )  ->  A  .|| 
B )
20 rhmdvdsr.m . . . . . . . 8  |-  .||  =  (
||r `  R )
213, 20, 14dvdsr2 17887 . . . . . . 7  |-  ( A  e.  X  ->  ( A  .||  B  <->  E. c  e.  X  ( c
( .r `  R
) A )  =  B ) )
2221biimpac 489 . . . . . 6  |-  ( ( A  .||  B  /\  A  e.  X )  ->  E. c  e.  X  ( c ( .r
`  R ) A )  =  B )
2319, 2, 22syl2anc 667 . . . . 5  |-  ( ( ( F  e.  ( R RingHom  S )  /\  A  e.  X  /\  B  e.  X )  /\  A  .|| 
B )  ->  E. c  e.  X  ( c
( .r `  R
) A )  =  B )
24 r19.29 2927 . . . . . 6  |-  ( ( A. c  e.  X  ( F `  ( c ( .r `  R
) A ) )  =  ( ( F `
 c ) ( .r `  S ) ( F `  A
) )  /\  E. c  e.  X  (
c ( .r `  R ) A )  =  B )  ->  E. c  e.  X  ( ( F `  ( c ( .r
`  R ) A ) )  =  ( ( F `  c
) ( .r `  S ) ( F `
 A ) )  /\  ( c ( .r `  R ) A )  =  B ) )
25 simpl 459 . . . . . . . 8  |-  ( ( ( F `  (
c ( .r `  R ) A ) )  =  ( ( F `  c ) ( .r `  S
) ( F `  A ) )  /\  ( c ( .r
`  R ) A )  =  B )  ->  ( F `  ( c ( .r
`  R ) A ) )  =  ( ( F `  c
) ( .r `  S ) ( F `
 A ) ) )
26 simpr 463 . . . . . . . . 9  |-  ( ( ( F `  (
c ( .r `  R ) A ) )  =  ( ( F `  c ) ( .r `  S
) ( F `  A ) )  /\  ( c ( .r
`  R ) A )  =  B )  ->  ( c ( .r `  R ) A )  =  B )
2726fveq2d 5874 . . . . . . . 8  |-  ( ( ( F `  (
c ( .r `  R ) A ) )  =  ( ( F `  c ) ( .r `  S
) ( F `  A ) )  /\  ( c ( .r
`  R ) A )  =  B )  ->  ( F `  ( c ( .r
`  R ) A ) )  =  ( F `  B ) )
2825, 27eqtr3d 2489 . . . . . . 7  |-  ( ( ( F `  (
c ( .r `  R ) A ) )  =  ( ( F `  c ) ( .r `  S
) ( F `  A ) )  /\  ( c ( .r
`  R ) A )  =  B )  ->  ( ( F `
 c ) ( .r `  S ) ( F `  A
) )  =  ( F `  B ) )
2928reximi 2857 . . . . . 6  |-  ( E. c  e.  X  ( ( F `  (
c ( .r `  R ) A ) )  =  ( ( F `  c ) ( .r `  S
) ( F `  A ) )  /\  ( c ( .r
`  R ) A )  =  B )  ->  E. c  e.  X  ( ( F `  c ) ( .r
`  S ) ( F `  A ) )  =  ( F `
 B ) )
3024, 29syl 17 . . . . 5  |-  ( ( A. c  e.  X  ( F `  ( c ( .r `  R
) A ) )  =  ( ( F `
 c ) ( .r `  S ) ( F `  A
) )  /\  E. c  e.  X  (
c ( .r `  R ) A )  =  B )  ->  E. c  e.  X  ( ( F `  c ) ( .r
`  S ) ( F `  A ) )  =  ( F `
 B ) )
3118, 23, 30syl2anc 667 . . . 4  |-  ( ( ( F  e.  ( R RingHom  S )  /\  A  e.  X  /\  B  e.  X )  /\  A  .|| 
B )  ->  E. c  e.  X  ( ( F `  c )
( .r `  S
) ( F `  A ) )  =  ( F `  B
) )
32 r19.29 2927 . . . 4  |-  ( ( A. c  e.  X  ( F `  c )  e.  ( Base `  S
)  /\  E. c  e.  X  ( ( F `  c )
( .r `  S
) ( F `  A ) )  =  ( F `  B
) )  ->  E. c  e.  X  ( ( F `  c )  e.  ( Base `  S
)  /\  ( ( F `  c )
( .r `  S
) ( F `  A ) )  =  ( F `  B
) ) )
3312, 31, 32syl2anc 667 . . 3  |-  ( ( ( F  e.  ( R RingHom  S )  /\  A  e.  X  /\  B  e.  X )  /\  A  .|| 
B )  ->  E. c  e.  X  ( ( F `  c )  e.  ( Base `  S
)  /\  ( ( F `  c )
( .r `  S
) ( F `  A ) )  =  ( F `  B
) ) )
34 oveq1 6302 . . . . . 6  |-  ( y  =  ( F `  c )  ->  (
y ( .r `  S ) ( F `
 A ) )  =  ( ( F `
 c ) ( .r `  S ) ( F `  A
) ) )
3534eqeq1d 2455 . . . . 5  |-  ( y  =  ( F `  c )  ->  (
( y ( .r
`  S ) ( F `  A ) )  =  ( F `
 B )  <->  ( ( F `  c )
( .r `  S
) ( F `  A ) )  =  ( F `  B
) ) )
3635rspcev 3152 . . . 4  |-  ( ( ( F `  c
)  e.  ( Base `  S )  /\  (
( F `  c
) ( .r `  S ) ( F `
 A ) )  =  ( F `  B ) )  ->  E. y  e.  ( Base `  S ) ( y ( .r `  S ) ( F `
 A ) )  =  ( F `  B ) )
3736rexlimivw 2878 . . 3  |-  ( E. c  e.  X  ( ( F `  c
)  e.  ( Base `  S )  /\  (
( F `  c
) ( .r `  S ) ( F `
 A ) )  =  ( F `  B ) )  ->  E. y  e.  ( Base `  S ) ( y ( .r `  S ) ( F `
 A ) )  =  ( F `  B ) )
3833, 37syl 17 . 2  |-  ( ( ( F  e.  ( R RingHom  S )  /\  A  e.  X  /\  B  e.  X )  /\  A  .|| 
B )  ->  E. y  e.  ( Base `  S
) ( y ( .r `  S ) ( F `  A
) )  =  ( F `  B ) )
39 rhmdvdsr.n . . 3  |-  ./  =  ( ||r `
 S )
404, 39, 15dvdsr 17886 . 2  |-  ( ( F `  A ) 
./  ( F `  B )  <->  ( ( F `  A )  e.  ( Base `  S
)  /\  E. y  e.  ( Base `  S
) ( y ( .r `  S ) ( F `  A
) )  =  ( F `  B ) ) )
417, 38, 40sylanbrc 671 1  |-  ( ( ( F  e.  ( R RingHom  S )  /\  A  e.  X  /\  B  e.  X )  /\  A  .|| 
B )  ->  ( F `  A )  ./  ( F `  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889   A.wral 2739   E.wrex 2740   class class class wbr 4405   ` cfv 5585  (class class class)co 6295   Basecbs 15133   .rcmulr 15203   ||rcdsr 17878   RingHom crh 17952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-ndx 15136  df-slot 15137  df-base 15138  df-sets 15139  df-plusg 15215  df-0g 15352  df-mhm 16594  df-ghm 16893  df-mgp 17736  df-ur 17748  df-ring 17794  df-dvdsr 17881  df-rnghom 17955
This theorem is referenced by:  elrhmunit  28595
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