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Theorem asclpropd 16085
 Description: If two structures have the same components (properties), one is an associative algebra iff the other one is. The last hypotheses on can be discharged either by letting (if strong equality is known on ) or assuming is a ring. (Contributed by Mario Carneiro, 5-Jul-2015.)
Hypotheses
Ref Expression
asclpropd.f Scalar
asclpropd.g Scalar
asclpropd.1
asclpropd.2
asclpropd.3
asclpropd.4
asclpropd.5
Assertion
Ref Expression
asclpropd algSc algSc
Distinct variable groups:   ,,   ,,   ,,   ,,   ,,
Allowed substitution hints:   (,)   (,)

Proof of Theorem asclpropd
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 asclpropd.5 . . . . . . 7
21adantr 451 . . . . . 6
3 asclpropd.3 . . . . . . . 8
43proplem 13592 . . . . . . 7
54anassrs 629 . . . . . 6
62, 5mpdan 649 . . . . 5
7 asclpropd.4 . . . . . . 7
87oveq2d 5874 . . . . . 6
98adantr 451 . . . . 5
106, 9eqtrd 2315 . . . 4
1110mpteq2dva 4106 . . 3
12 asclpropd.1 . . . 4
13 mpteq1 4100 . . . 4
1412, 13syl 15 . . 3
15 asclpropd.2 . . . 4
16 mpteq1 4100 . . . 4
1715, 16syl 15 . . 3
1811, 14, 173eqtr3d 2323 . 2
19 eqid 2283 . . 3 algSc algSc
20 asclpropd.f . . 3 Scalar
21 eqid 2283 . . 3
22 eqid 2283 . . 3
23 eqid 2283 . . 3
2419, 20, 21, 22, 23asclfval 16074 . 2 algSc
25 eqid 2283 . . 3 algSc algSc
26 asclpropd.g . . 3 Scalar
27 eqid 2283 . . 3
28 eqid 2283 . . 3
29 eqid 2283 . . 3
3025, 26, 27, 28, 29asclfval 16074 . 2 algSc
3118, 24, 303eqtr4g 2340 1 algSc algSc
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   wceq 1623   wcel 1684   cmpt 4077  cfv 5255  (class class class)co 5858  cbs 13148  Scalarcsca 13211  cvsca 13212  cur 15339  algSccascl 16052 This theorem is referenced by:  ply1ascl  16335 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-slot 13152  df-base 13153  df-ascl 16055
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