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Theorem toycom 37832
Description: Show the commutative law for an operation 𝑂 on a toy structure class 𝐶 of commuatitive operations on . This illustrates how a structure class can be partially specialized. In practice, we would ordinarily define a new constant such as "CAbel" in place of 𝐶. (Contributed by NM, 17-Mar-2013.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
toycom.1 𝐶 = {𝑔 ∈ Abel ∣ (Base‘𝑔) = ℂ}
toycom.2 + = (+g𝐾)
Assertion
Ref Expression
toycom ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
Distinct variable group:   𝑔,𝐾
Allowed substitution hints:   𝐴(𝑔)   𝐵(𝑔)   𝐶(𝑔)   + (𝑔)

Proof of Theorem toycom
StepHypRef Expression
1 toycom.1 . . . . . 6 𝐶 = {𝑔 ∈ Abel ∣ (Base‘𝑔) = ℂ}
2 ssrab2 4077 . . . . . 6 {𝑔 ∈ Abel ∣ (Base‘𝑔) = ℂ} ⊆ Abel
31, 2eqsstri 4016 . . . . 5 𝐶 ⊆ Abel
43sseli 3978 . . . 4 (𝐾𝐶𝐾 ∈ Abel)
543ad2ant1 1134 . . 3 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐾 ∈ Abel)
6 simp2 1138 . . . 4 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ ℂ)
7 fveq2 6889 . . . . . . . 8 (𝑔 = 𝐾 → (Base‘𝑔) = (Base‘𝐾))
87eqeq1d 2735 . . . . . . 7 (𝑔 = 𝐾 → ((Base‘𝑔) = ℂ ↔ (Base‘𝐾) = ℂ))
98, 1elrab2 3686 . . . . . 6 (𝐾𝐶 ↔ (𝐾 ∈ Abel ∧ (Base‘𝐾) = ℂ))
109simprbi 498 . . . . 5 (𝐾𝐶 → (Base‘𝐾) = ℂ)
11103ad2ant1 1134 . . . 4 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (Base‘𝐾) = ℂ)
126, 11eleqtrrd 2837 . . 3 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ (Base‘𝐾))
13 simp3 1139 . . . 4 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ)
1413, 11eleqtrrd 2837 . . 3 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ (Base‘𝐾))
15 eqid 2733 . . . 4 (Base‘𝐾) = (Base‘𝐾)
16 eqid 2733 . . . 4 (+g𝐾) = (+g𝐾)
1715, 16ablcom 19662 . . 3 ((𝐾 ∈ Abel ∧ 𝐴 ∈ (Base‘𝐾) ∧ 𝐵 ∈ (Base‘𝐾)) → (𝐴(+g𝐾)𝐵) = (𝐵(+g𝐾)𝐴))
185, 12, 14, 17syl3anc 1372 . 2 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴(+g𝐾)𝐵) = (𝐵(+g𝐾)𝐴))
19 toycom.2 . . 3 + = (+g𝐾)
2019oveqi 7419 . 2 (𝐴 + 𝐵) = (𝐴(+g𝐾)𝐵)
2119oveqi 7419 . 2 (𝐵 + 𝐴) = (𝐵(+g𝐾)𝐴)
2218, 20, 213eqtr4g 2798 1 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088   = wceq 1542  wcel 2107  {crab 3433  cfv 6541  (class class class)co 7406  cc 11105  Basecbs 17141  +gcplusg 17194  Abelcabl 19644
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-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6493  df-fv 6549  df-ov 7409  df-cmn 19645  df-abl 19646
This theorem is referenced by: (None)
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