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Theorem rhmdvd 26224
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 983 . . . 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 1016 . . . 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 1018 . . . 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 2441 . . . . 5  |-  ( .r
`  S )  =  ( .r `  S
)
74, 5, 6rhmmul 16805 . . . 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 1213 . . 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 1017 . . . 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 16805 . . . 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 1213 . . 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 6108 . 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 16800 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  S  e.  Ring )
14133ad2ant1 1004 . . 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 2441 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
164, 15rhmf 16804 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  F : X
--> ( Base `  S
) )
17163ad2ant1 1004 . . . 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 5841 . . 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 1011 . . 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 1012 . . 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 26196 . . 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 1215 . 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 2474 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 960    = wceq 1364    e. wcel 1761   -->wf 5411   ` cfv 5415  (class class class)co 6090   Basecbs 14170   .rcmulr 14235   Ringcrg 16635  Unitcui 16721  /rcdvr 16764   RingHom crh 16794
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-recs 6828  df-rdg 6862  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-0g 14376  df-mnd 15411  df-mhm 15460  df-grp 15538  df-minusg 15539  df-ghm 15738  df-mgp 16582  df-ur 16594  df-rng 16637  df-oppr 16705  df-dvdsr 16723  df-unit 16724  df-invr 16754  df-dvr 16765  df-rnghom 16796
This theorem is referenced by:  qqhval2lem  26346  qqhghm  26353  qqhrhm  26354
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