Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rhmdvd Structured version   Unicode version

Theorem rhmdvd 26429
Description: A ring homomorphism preserves ratios. (Contributed by Thierry Arnoux, 22-Oct-2017.)
Hypotheses
Ref Expression
rhmdvd.u  |-  U  =  (Unit `  S )
rhmdvd.x  |-  X  =  ( Base `  R
)
rhmdvd.d  |-  ./  =  (/r
`  S )
rhmdvd.m  |-  .x.  =  ( .r `  R )
Assertion
Ref Expression
rhmdvd  |-  ( ( F  e.  ( R RingHom  S )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  /\  ( ( F `  B )  e.  U  /\  ( F `  C
)  e.  U ) )  ->  ( ( F `  A )  ./  ( F `  B
) )  =  ( ( F `  ( A  .x.  C ) ) 
./  ( F `  ( B  .x.  C ) ) ) )

Proof of Theorem rhmdvd
StepHypRef Expression
1 simp1 988 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  /\  ( ( F `  B )  e.  U  /\  ( F `  C
)  e.  U ) )  ->  F  e.  ( R RingHom  S ) )
2 simp21 1021 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  /\  ( ( F `  B )  e.  U  /\  ( F `  C
)  e.  U ) )  ->  A  e.  X )
3 simp23 1023 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  /\  ( ( F `  B )  e.  U  /\  ( F `  C
)  e.  U ) )  ->  C  e.  X )
4 rhmdvd.x . . . . 5  |-  X  =  ( Base `  R
)
5 rhmdvd.m . . . . 5  |-  .x.  =  ( .r `  R )
6 eqid 2452 . . . . 5  |-  ( .r
`  S )  =  ( .r `  S
)
74, 5, 6rhmmul 16935 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  X  /\  C  e.  X )  ->  ( F `  ( A  .x.  C ) )  =  ( ( F `  A ) ( .r
`  S ) ( F `  C ) ) )
81, 2, 3, 7syl3anc 1219 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  /\  ( ( F `  B )  e.  U  /\  ( F `  C
)  e.  U ) )  ->  ( F `  ( A  .x.  C
) )  =  ( ( F `  A
) ( .r `  S ) ( F `
 C ) ) )
9 simp22 1022 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  /\  ( ( F `  B )  e.  U  /\  ( F `  C
)  e.  U ) )  ->  B  e.  X )
104, 5, 6rhmmul 16935 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  B  e.  X  /\  C  e.  X )  ->  ( F `  ( B  .x.  C ) )  =  ( ( F `  B ) ( .r
`  S ) ( F `  C ) ) )
111, 9, 3, 10syl3anc 1219 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  /\  ( ( F `  B )  e.  U  /\  ( F `  C
)  e.  U ) )  ->  ( F `  ( B  .x.  C
) )  =  ( ( F `  B
) ( .r `  S ) ( F `
 C ) ) )
128, 11oveq12d 6213 . 2  |-  ( ( F  e.  ( R RingHom  S )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  /\  ( ( F `  B )  e.  U  /\  ( F `  C
)  e.  U ) )  ->  ( ( F `  ( A  .x.  C ) )  ./  ( F `  ( B 
.x.  C ) ) )  =  ( ( ( F `  A
) ( .r `  S ) ( F `
 C ) ) 
./  ( ( F `
 B ) ( .r `  S ) ( F `  C
) ) ) )
13 rhmrcl2 16928 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  S  e.  Ring )
14133ad2ant1 1009 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  /\  ( ( F `  B )  e.  U  /\  ( F `  C
)  e.  U ) )  ->  S  e.  Ring )
15 eqid 2452 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
164, 15rhmf 16934 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  F : X
--> ( Base `  S
) )
17163ad2ant1 1009 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  /\  ( ( F `  B )  e.  U  /\  ( F `  C
)  e.  U ) )  ->  F : X
--> ( Base `  S
) )
1817, 2ffvelrnd 5948 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  /\  ( ( F `  B )  e.  U  /\  ( F `  C
)  e.  U ) )  ->  ( F `  A )  e.  (
Base `  S )
)
19 simp3l 1016 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  /\  ( ( F `  B )  e.  U  /\  ( F `  C
)  e.  U ) )  ->  ( F `  B )  e.  U
)
20 simp3r 1017 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  /\  ( ( F `  B )  e.  U  /\  ( F `  C
)  e.  U ) )  ->  ( F `  C )  e.  U
)
21 rhmdvd.u . . . 4  |-  U  =  (Unit `  S )
22 rhmdvd.d . . . 4  |-  ./  =  (/r
`  S )
2315, 21, 22, 6dvrcan5 26401 . . 3  |-  ( ( S  e.  Ring  /\  (
( F `  A
)  e.  ( Base `  S )  /\  ( F `  B )  e.  U  /\  ( F `  C )  e.  U ) )  -> 
( ( ( F `
 A ) ( .r `  S ) ( F `  C
) )  ./  (
( F `  B
) ( .r `  S ) ( F `
 C ) ) )  =  ( ( F `  A ) 
./  ( F `  B ) ) )
2414, 18, 19, 20, 23syl13anc 1221 . 2  |-  ( ( F  e.  ( R RingHom  S )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  /\  ( ( F `  B )  e.  U  /\  ( F `  C
)  e.  U ) )  ->  ( (
( F `  A
) ( .r `  S ) ( F `
 C ) ) 
./  ( ( F `
 B ) ( .r `  S ) ( F `  C
) ) )  =  ( ( F `  A )  ./  ( F `  B )
) )
2512, 24eqtr2d 2494 1  |-  ( ( F  e.  ( R RingHom  S )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  /\  ( ( F `  B )  e.  U  /\  ( F `  C
)  e.  U ) )  ->  ( ( F `  A )  ./  ( F `  B
) )  =  ( ( F `  ( A  .x.  C ) ) 
./  ( F `  ( B  .x.  C ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   -->wf 5517   ` cfv 5521  (class class class)co 6195   Basecbs 14287   .rcmulr 14353   Ringcrg 16763  Unitcui 16849  /rcdvr 16892   RingHom crh 16922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-tpos 6850  df-recs 6937  df-rdg 6971  df-er 7206  df-map 7321  df-en 7416  df-dom 7417  df-sdom 7418  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-2 10486  df-3 10487  df-ndx 14290  df-slot 14291  df-base 14292  df-sets 14293  df-ress 14294  df-plusg 14365  df-mulr 14366  df-0g 14494  df-mnd 15529  df-mhm 15578  df-grp 15659  df-minusg 15660  df-ghm 15859  df-mgp 16709  df-ur 16721  df-rng 16765  df-oppr 16833  df-dvdsr 16851  df-unit 16852  df-invr 16882  df-dvr 16893  df-rnghom 16924
This theorem is referenced by:  qqhval2lem  26550  qqhghm  26557  qqhrhm  26558
  Copyright terms: Public domain W3C validator