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Theorem asclpropd 16396
 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 452 . . . . . 6
3 asclpropd.3 . . . . . . . 8
43proplem 13907 . . . . . . 7
54anassrs 630 . . . . . 6
62, 5mpdan 650 . . . . 5
7 asclpropd.4 . . . . . . 7
87oveq2d 6089 . . . . . 6
98adantr 452 . . . . 5
106, 9eqtrd 2467 . . . 4
1110mpteq2dva 4287 . . 3
12 asclpropd.1 . . . 4
1312mpteq1d 4282 . . 3
14 asclpropd.2 . . . 4
1514mpteq1d 4282 . . 3
1611, 13, 153eqtr3d 2475 . 2
17 eqid 2435 . . 3 algSc algSc
18 asclpropd.f . . 3 Scalar
19 eqid 2435 . . 3
20 eqid 2435 . . 3
21 eqid 2435 . . 3
2217, 18, 19, 20, 21asclfval 16385 . 2 algSc
23 eqid 2435 . . 3 algSc algSc
24 asclpropd.g . . 3 Scalar
25 eqid 2435 . . 3
26 eqid 2435 . . 3
27 eqid 2435 . . 3
2823, 24, 25, 26, 27asclfval 16385 . 2 algSc
2916, 22, 283eqtr4g 2492 1 algSc algSc
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725   cmpt 4258  cfv 5446  (class class class)co 6073  cbs 13461  Scalarcsca 13524  cvsca 13525  cur 15654  algSccascl 16363 This theorem is referenced by:  ply1ascl  16643 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-slot 13465  df-base 13466  df-ascl 16366
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