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Theorem asclpropd 18205
 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 463 . . . . . 6
3 asclpropd.3 . . . . . . . 8
43oveqrspc2v 6255 . . . . . . 7
54anassrs 646 . . . . . 6
62, 5mpdan 666 . . . . 5
7 asclpropd.4 . . . . . . 7
87oveq2d 6248 . . . . . 6
98adantr 463 . . . . 5
106, 9eqtrd 2441 . . . 4
1110mpteq2dva 4478 . . 3
12 asclpropd.1 . . . 4
1312mpteq1d 4473 . . 3
14 asclpropd.2 . . . 4
1514mpteq1d 4473 . . 3
1611, 13, 153eqtr3d 2449 . 2
17 eqid 2400 . . 3 algSc algSc
18 asclpropd.f . . 3 Scalar
19 eqid 2400 . . 3
20 eqid 2400 . . 3
21 eqid 2400 . . 3
2217, 18, 19, 20, 21asclfval 18193 . 2 algSc
23 eqid 2400 . . 3 algSc algSc
24 asclpropd.g . . 3 Scalar
25 eqid 2400 . . 3
26 eqid 2400 . . 3
27 eqid 2400 . . 3
2823, 24, 25, 26, 27asclfval 18193 . 2 algSc
2916, 22, 283eqtr4g 2466 1 algSc algSc
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 367   wceq 1403   wcel 1840   cmpt 4450  cfv 5523  (class class class)co 6232  cbs 14731  Scalarcsca 14802  cvsca 14803  cur 17363  algSccascl 18170 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-ov 6235  df-slot 14735  df-base 14736  df-ascl 18173 This theorem is referenced by:  ply1ascl  18509
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