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Theorem prsubrtr 25502
 Description: The product of a subset of by an element of is the image of by a right translation. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)
Hypotheses
Ref Expression
trfun.2
trinv.1
prsubrtr.1
Assertion
Ref Expression
prsubrtr
Distinct variable groups:   ,   ,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem prsubrtr
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5882 . . . . . . 7
21eqeq2d 2307 . . . . . 6
32rexsng 3686 . . . . 5
43rexbidv 2577 . . . 4
6 simp1 955 . . . 4
7 trinv.1 . . . . . . . . 9
8 grporndm 20893 . . . . . . . . 9
97, 8syl5eq 2340 . . . . . . . 8
109pweqd 3643 . . . . . . 7
1110eleq2d 2363 . . . . . 6
1211biimpa 470 . . . . 5
13123adant2 974 . . . 4
149eleq2d 2363 . . . . . . 7
1514biimpa 470 . . . . . 6
16153adant3 975 . . . . 5
17 snelpwi 4236 . . . . 5
1816, 17syl 15 . . . 4
19 eqid 2296 . . . . 5
20 prsubrtr.1 . . . . 5
2119, 20iscst3 25279 . . . 4
226, 13, 18, 21syl3anc 1182 . . 3
23 df-ima 4718 . . . . . 6
24 trfun.2 . . . . . . . . 9
2524reseq1i 4967 . . . . . . . 8
26 elpwi 3646 . . . . . . . . . 10
27263ad2ant3 978 . . . . . . . . 9
28 resmpt 5016 . . . . . . . . 9
2927, 28syl 15 . . . . . . . 8
3025, 29syl5eq 2340 . . . . . . 7
3130rneqd 4922 . . . . . 6
3223, 31syl5eq 2340 . . . . 5
3332eleq2d 2363 . . . 4
34 eqid 2296 . . . . 5
35 ovex 5899 . . . . 5
3634, 35elrnmpti 4946 . . . 4
3733, 36syl6bb 252 . . 3
385, 22, 373bitr4d 276 . 2
3938eqrdv 2294 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   w3a 934   wceq 1632   wcel 1696  wrex 2557   wss 3165  cpw 3638  csn 3653   cmpt 4093   cdm 4705   crn 4706   cres 4707  cima 4708  cfv 5271  (class class class)co 5874  cgr 20869  ccst 25275 This theorem is referenced by:  caytr  25503 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-grpo 20874  df-cst 25276
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