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Theorem rhmdvd 28049
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 994 . . . 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 1027 . . . 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 1029 . . . 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 2454 . . . . 5  |-  ( .r
`  S )  =  ( .r `  S
)
74, 5, 6rhmmul 17574 . . . 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 1226 . . 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 1028 . . . 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 17574 . . . 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 1226 . . 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 6288 . 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 17567 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  S  e.  Ring )
14133ad2ant1 1015 . . 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 2454 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
164, 15rhmf 17573 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  F : X
--> ( Base `  S
) )
17163ad2ant1 1015 . . . 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 6008 . . 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 1022 . . 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 1023 . . 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 28021 . . 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 1228 . 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 2496 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   -->wf 5566   ` cfv 5570  (class class class)co 6270   Basecbs 14719   .rcmulr 14788   Ringcrg 17396  Unitcui 17486  /rcdvr 17529   RingHom crh 17559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-tpos 6947  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-mulr 14801  df-0g 14934  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-mhm 16168  df-grp 16259  df-minusg 16260  df-ghm 16467  df-mgp 17340  df-ur 17352  df-ring 17398  df-oppr 17470  df-dvdsr 17488  df-unit 17489  df-invr 17519  df-dvr 17530  df-rnghom 17562
This theorem is referenced by:  qqhval2lem  28199  qqhghm  28206  qqhrhm  28207
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