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Theorem toycom 33278
 Description: Show the commutative law for an operation 𝑂 on a toy structure class 𝐶 of commuatitive operations on ℂ. 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 3650 . . . . . 6 {𝑔 ∈ Abel ∣ (Base‘𝑔) = ℂ} ⊆ Abel
31, 2eqsstri 3598 . . . . 5 𝐶 ⊆ Abel
43sseli 3564 . . . 4 (𝐾𝐶𝐾 ∈ Abel)
543ad2ant1 1075 . . 3 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐾 ∈ Abel)
6 simp2 1055 . . . 4 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ ℂ)
7 fveq2 6103 . . . . . . . 8 (𝑔 = 𝐾 → (Base‘𝑔) = (Base‘𝐾))
87eqeq1d 2612 . . . . . . 7 (𝑔 = 𝐾 → ((Base‘𝑔) = ℂ ↔ (Base‘𝐾) = ℂ))
98, 1elrab2 3333 . . . . . 6 (𝐾𝐶 ↔ (𝐾 ∈ Abel ∧ (Base‘𝐾) = ℂ))
109simprbi 479 . . . . 5 (𝐾𝐶 → (Base‘𝐾) = ℂ)
11103ad2ant1 1075 . . . 4 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (Base‘𝐾) = ℂ)
126, 11eleqtrrd 2691 . . 3 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ (Base‘𝐾))
13 simp3 1056 . . . 4 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ)
1413, 11eleqtrrd 2691 . . 3 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ (Base‘𝐾))
15 eqid 2610 . . . 4 (Base‘𝐾) = (Base‘𝐾)
16 eqid 2610 . . . 4 (+g𝐾) = (+g𝐾)
1715, 16ablcom 18033 . . 3 ((𝐾 ∈ Abel ∧ 𝐴 ∈ (Base‘𝐾) ∧ 𝐵 ∈ (Base‘𝐾)) → (𝐴(+g𝐾)𝐵) = (𝐵(+g𝐾)𝐴))
185, 12, 14, 17syl3anc 1318 . 2 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴(+g𝐾)𝐵) = (𝐵(+g𝐾)𝐴))
19 toycom.2 . . 3 + = (+g𝐾)
2019oveqi 6562 . 2 (𝐴 + 𝐵) = (𝐴(+g𝐾)𝐵)
2119oveqi 6562 . 2 (𝐵 + 𝐴) = (𝐵(+g𝐾)𝐴)
2218, 20, 213eqtr4g 2669 1 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  Basecbs 15695  +gcplusg 15768  Abelcabl 18017 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-cmn 18018  df-abl 18019 This theorem is referenced by: (None)
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