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Theorem rhmdvdsr 27459
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 994 . . 3  |-  ( ( ( F  e.  ( R RingHom  S )  /\  A  e.  X  /\  B  e.  X )  /\  A  .|| 
B )  ->  F  e.  ( R RingHom  S )
)
2 simpl2 995 . . 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 2462 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
53, 4rhmf 17154 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  F : X
--> ( Base `  S
) )
65ffvelrnda 6014 . . 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 1030 . . . . . 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 6014 . . . . . 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 2873 . . . 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 2462 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
15 eqid 2462 . . . . . . . 8  |-  ( .r
`  S )  =  ( .r `  S
)
163, 14, 15rhmmul 17155 . . . . . . 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 1223 . . . . . 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 2873 . . . . 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 17075 . . . . . . 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 5863 . . . . . . . 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 2505 . . . . . . 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 2927 . . . . . 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 6284 . . . . . 6  |-  ( y  =  ( F `  c )  ->  (
y ( .r `  S ) ( F `
 A ) )  =  ( ( F `
 c ) ( .r `  S ) ( F `  A
) ) )
3534eqeq1d 2464 . . . . 5  |-  ( y  =  ( F `  c )  ->  (
( y ( .r
`  S ) ( F `  A ) )  =  ( F `
 B )  <->  ( ( F `  c )
( .r `  S
) ( F `  A ) )  =  ( F `  B
) ) )
3635rspcev 3209 . . . 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 2947 . . 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 17074 . 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 968    = wceq 1374    e. wcel 1762   A.wral 2809   E.wrex 2810   class class class wbr 4442   ` cfv 5581  (class class class)co 6277   Basecbs 14481   .rcmulr 14547   ||rcdsr 17066   RingHom crh 17140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-plusg 14559  df-0g 14688  df-mhm 15772  df-ghm 16055  df-mgp 16927  df-ur 16939  df-rng 16983  df-dvdsr 17069  df-rnghom 17143
This theorem is referenced by:  elrhmunit  27461
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