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Theorem asclpropd 21316
Description: If two structures have the same components (properties), one is an associative algebra iff the other one is. The last hypotheses on 1r can be discharged either by letting π‘Š = V (if strong equality is known on ·𝑠) or assuming 𝐾 is a ring. (Contributed by Mario Carneiro, 5-Jul-2015.)
Hypotheses
Ref Expression
asclpropd.f 𝐹 = (Scalarβ€˜πΎ)
asclpropd.g 𝐺 = (Scalarβ€˜πΏ)
asclpropd.1 (πœ‘ β†’ 𝑃 = (Baseβ€˜πΉ))
asclpropd.2 (πœ‘ β†’ 𝑃 = (Baseβ€˜πΊ))
asclpropd.3 ((πœ‘ ∧ (π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ π‘Š)) β†’ (π‘₯( ·𝑠 β€˜πΎ)𝑦) = (π‘₯( ·𝑠 β€˜πΏ)𝑦))
asclpropd.4 (πœ‘ β†’ (1rβ€˜πΎ) = (1rβ€˜πΏ))
asclpropd.5 (πœ‘ β†’ (1rβ€˜πΎ) ∈ π‘Š)
Assertion
Ref Expression
asclpropd (πœ‘ β†’ (algScβ€˜πΎ) = (algScβ€˜πΏ))
Distinct variable groups:   π‘₯,𝑦,𝐾   π‘₯,𝐿,𝑦   π‘₯,𝑃,𝑦   πœ‘,π‘₯,𝑦   π‘₯,π‘Š,𝑦
Allowed substitution hints:   𝐹(π‘₯,𝑦)   𝐺(π‘₯,𝑦)

Proof of Theorem asclpropd
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 asclpropd.5 . . . . . 6 (πœ‘ β†’ (1rβ€˜πΎ) ∈ π‘Š)
2 asclpropd.3 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ π‘Š)) β†’ (π‘₯( ·𝑠 β€˜πΎ)𝑦) = (π‘₯( ·𝑠 β€˜πΏ)𝑦))
32oveqrspc2v 7385 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ 𝑃 ∧ (1rβ€˜πΎ) ∈ π‘Š)) β†’ (𝑧( ·𝑠 β€˜πΎ)(1rβ€˜πΎ)) = (𝑧( ·𝑠 β€˜πΏ)(1rβ€˜πΎ)))
43anassrs 469 . . . . . 6 (((πœ‘ ∧ 𝑧 ∈ 𝑃) ∧ (1rβ€˜πΎ) ∈ π‘Š) β†’ (𝑧( ·𝑠 β€˜πΎ)(1rβ€˜πΎ)) = (𝑧( ·𝑠 β€˜πΏ)(1rβ€˜πΎ)))
51, 4mpidan 688 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ 𝑃) β†’ (𝑧( ·𝑠 β€˜πΎ)(1rβ€˜πΎ)) = (𝑧( ·𝑠 β€˜πΏ)(1rβ€˜πΎ)))
6 asclpropd.4 . . . . . . 7 (πœ‘ β†’ (1rβ€˜πΎ) = (1rβ€˜πΏ))
76oveq2d 7374 . . . . . 6 (πœ‘ β†’ (𝑧( ·𝑠 β€˜πΏ)(1rβ€˜πΎ)) = (𝑧( ·𝑠 β€˜πΏ)(1rβ€˜πΏ)))
87adantr 482 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ 𝑃) β†’ (𝑧( ·𝑠 β€˜πΏ)(1rβ€˜πΎ)) = (𝑧( ·𝑠 β€˜πΏ)(1rβ€˜πΏ)))
95, 8eqtrd 2773 . . . 4 ((πœ‘ ∧ 𝑧 ∈ 𝑃) β†’ (𝑧( ·𝑠 β€˜πΎ)(1rβ€˜πΎ)) = (𝑧( ·𝑠 β€˜πΏ)(1rβ€˜πΏ)))
109mpteq2dva 5206 . . 3 (πœ‘ β†’ (𝑧 ∈ 𝑃 ↦ (𝑧( ·𝑠 β€˜πΎ)(1rβ€˜πΎ))) = (𝑧 ∈ 𝑃 ↦ (𝑧( ·𝑠 β€˜πΏ)(1rβ€˜πΏ))))
11 asclpropd.1 . . . 4 (πœ‘ β†’ 𝑃 = (Baseβ€˜πΉ))
1211mpteq1d 5201 . . 3 (πœ‘ β†’ (𝑧 ∈ 𝑃 ↦ (𝑧( ·𝑠 β€˜πΎ)(1rβ€˜πΎ))) = (𝑧 ∈ (Baseβ€˜πΉ) ↦ (𝑧( ·𝑠 β€˜πΎ)(1rβ€˜πΎ))))
13 asclpropd.2 . . . 4 (πœ‘ β†’ 𝑃 = (Baseβ€˜πΊ))
1413mpteq1d 5201 . . 3 (πœ‘ β†’ (𝑧 ∈ 𝑃 ↦ (𝑧( ·𝑠 β€˜πΏ)(1rβ€˜πΏ))) = (𝑧 ∈ (Baseβ€˜πΊ) ↦ (𝑧( ·𝑠 β€˜πΏ)(1rβ€˜πΏ))))
1510, 12, 143eqtr3d 2781 . 2 (πœ‘ β†’ (𝑧 ∈ (Baseβ€˜πΉ) ↦ (𝑧( ·𝑠 β€˜πΎ)(1rβ€˜πΎ))) = (𝑧 ∈ (Baseβ€˜πΊ) ↦ (𝑧( ·𝑠 β€˜πΏ)(1rβ€˜πΏ))))
16 eqid 2733 . . 3 (algScβ€˜πΎ) = (algScβ€˜πΎ)
17 asclpropd.f . . 3 𝐹 = (Scalarβ€˜πΎ)
18 eqid 2733 . . 3 (Baseβ€˜πΉ) = (Baseβ€˜πΉ)
19 eqid 2733 . . 3 ( ·𝑠 β€˜πΎ) = ( ·𝑠 β€˜πΎ)
20 eqid 2733 . . 3 (1rβ€˜πΎ) = (1rβ€˜πΎ)
2116, 17, 18, 19, 20asclfval 21298 . 2 (algScβ€˜πΎ) = (𝑧 ∈ (Baseβ€˜πΉ) ↦ (𝑧( ·𝑠 β€˜πΎ)(1rβ€˜πΎ)))
22 eqid 2733 . . 3 (algScβ€˜πΏ) = (algScβ€˜πΏ)
23 asclpropd.g . . 3 𝐺 = (Scalarβ€˜πΏ)
24 eqid 2733 . . 3 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
25 eqid 2733 . . 3 ( ·𝑠 β€˜πΏ) = ( ·𝑠 β€˜πΏ)
26 eqid 2733 . . 3 (1rβ€˜πΏ) = (1rβ€˜πΏ)
2722, 23, 24, 25, 26asclfval 21298 . 2 (algScβ€˜πΏ) = (𝑧 ∈ (Baseβ€˜πΊ) ↦ (𝑧( ·𝑠 β€˜πΏ)(1rβ€˜πΏ)))
2815, 21, 273eqtr4g 2798 1 (πœ‘ β†’ (algScβ€˜πΎ) = (algScβ€˜πΏ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ↦ cmpt 5189  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  Scalarcsca 17141   ·𝑠 cvsca 17142  1rcur 19918  algSccascl 21274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-1cn 11114  ax-addcl 11116
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-om 7804  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-nn 12159  df-slot 17059  df-ndx 17071  df-base 17089  df-ascl 21277
This theorem is referenced by:  ply1ascl  21645
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