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Theorem rhmdvdsr 27969
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 999 . . 3  |-  ( ( ( F  e.  ( R RingHom  S )  /\  A  e.  X  /\  B  e.  X )  /\  A  .|| 
B )  ->  F  e.  ( R RingHom  S )
)
2 simpl2 1000 . . 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 2457 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
53, 4rhmf 17502 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  F : X
--> ( Base `  S
) )
65ffvelrnda 6032 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  X )  ->  ( F `  A )  e.  ( Base `  S
) )
71, 2, 6syl2anc 661 . 2  |-  ( ( ( F  e.  ( R RingHom  S )  /\  A  e.  X  /\  B  e.  X )  /\  A  .|| 
B )  ->  ( F `  A )  e.  ( Base `  S
) )
8 simpll1 1035 . . . . . 6  |-  ( ( ( ( F  e.  ( R RingHom  S )  /\  A  e.  X  /\  B  e.  X
)  /\  A  .||  B )  /\  c  e.  X
)  ->  F  e.  ( R RingHom  S ) )
9 simpr 461 . . . . . 6  |-  ( ( ( ( F  e.  ( R RingHom  S )  /\  A  e.  X  /\  B  e.  X
)  /\  A  .||  B )  /\  c  e.  X
)  ->  c  e.  X )
105ffvelrnda 6032 . . . . . 6  |-  ( ( F  e.  ( R RingHom  S )  /\  c  e.  X )  ->  ( F `  c )  e.  ( Base `  S
) )
118, 9, 10syl2anc 661 . . . . 5  |-  ( ( ( ( F  e.  ( R RingHom  S )  /\  A  e.  X  /\  B  e.  X
)  /\  A  .||  B )  /\  c  e.  X
)  ->  ( F `  c )  e.  (
Base `  S )
)
1211ralrimiva 2871 . . . 4  |-  ( ( ( F  e.  ( R RingHom  S )  /\  A  e.  X  /\  B  e.  X )  /\  A  .|| 
B )  ->  A. c  e.  X  ( F `  c )  e.  (
Base `  S )
)
132adantr 465 . . . . . . 7  |-  ( ( ( ( F  e.  ( R RingHom  S )  /\  A  e.  X  /\  B  e.  X
)  /\  A  .||  B )  /\  c  e.  X
)  ->  A  e.  X )
14 eqid 2457 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
15 eqid 2457 . . . . . . . 8  |-  ( .r
`  S )  =  ( .r `  S
)
163, 14, 15rhmmul 17503 . . . . . . 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 1228 . . . . . 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 2871 . . . . 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 461 . . . . . 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 17423 . . . . . . 7  |-  ( A  e.  X  ->  ( A  .||  B  <->  E. c  e.  X  ( c
( .r `  R
) A )  =  B ) )
2221biimpac 486 . . . . . 6  |-  ( ( A  .||  B  /\  A  e.  X )  ->  E. c  e.  X  ( c ( .r
`  R ) A )  =  B )
2319, 2, 22syl2anc 661 . . . . 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 2992 . . . . . 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 457 . . . . . . . 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 461 . . . . . . . . 9  |-  ( ( ( F `  (
c ( .r `  R ) A ) )  =  ( ( F `  c ) ( .r `  S
) ( F `  A ) )  /\  ( c ( .r
`  R ) A )  =  B )  ->  ( c ( .r `  R ) A )  =  B )
2726fveq2d 5876 . . . . . . . 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 2500 . . . . . . 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 2925 . . . . . 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 16 . . . . 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 661 . . . 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 2992 . . . 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 661 . . 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 6303 . . . . . 6  |-  ( y  =  ( F `  c )  ->  (
y ( .r `  S ) ( F `
 A ) )  =  ( ( F `
 c ) ( .r `  S ) ( F `  A
) ) )
3534eqeq1d 2459 . . . . 5  |-  ( y  =  ( F `  c )  ->  (
( y ( .r
`  S ) ( F `  A ) )  =  ( F `
 B )  <->  ( ( F `  c )
( .r `  S
) ( F `  A ) )  =  ( F `  B
) ) )
3635rspcev 3210 . . . 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 2946 . . 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 16 . 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 17422 . 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 664 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 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   Basecbs 14644   .rcmulr 14713   ||rcdsr 17414   RingHom crh 17488
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-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
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-nel 2655  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-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-plusg 14725  df-0g 14859  df-mhm 16093  df-ghm 16392  df-mgp 17269  df-ur 17281  df-ring 17327  df-dvdsr 17417  df-rnghom 17491
This theorem is referenced by:  elrhmunit  27971
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