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Theorem rhmdvd 24212
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 957 . . . 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 990 . . . 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 992 . . . 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 2404 . . . . 5  |-  ( .r
`  S )  =  ( .r `  S
)
74, 5, 6rhmmul 15783 . . . 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 1184 . . 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 991 . . . 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 15783 . . . 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 1184 . . 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 6058 . 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 15778 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  S  e.  Ring )
14133ad2ant1 978 . . 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 2404 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
164, 15rhmf 15782 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  F : X
--> ( Base `  S
) )
17163ad2ant1 978 . . . 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 5830 . . 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 985 . . 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 986 . . 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 24182 . . 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 1186 . 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 2437 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   -->wf 5409   ` cfv 5413  (class class class)co 6040   Basecbs 13424   .rcmulr 13485   Ringcrg 15615  Unitcui 15699  /rcdvr 15742   RingHom crh 15772
This theorem is referenced by:  qqhval2lem  24318  qqhghm  24325  qqhrhm  24326
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  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-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-0g 13682  df-mnd 14645  df-mhm 14693  df-grp 14767  df-minusg 14768  df-ghm 14959  df-mgp 15604  df-rng 15618  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-dvr 15743  df-rnghom 15774
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