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Theorem toycom 38955
Description: Show the commutative law for an operation 𝑂 on a toy structure class 𝐶 of commutative 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 4090 . . . . . 6 {𝑔 ∈ Abel ∣ (Base‘𝑔) = ℂ} ⊆ Abel
31, 2eqsstri 4030 . . . . 5 𝐶 ⊆ Abel
43sseli 3991 . . . 4 (𝐾𝐶𝐾 ∈ Abel)
543ad2ant1 1132 . . 3 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐾 ∈ Abel)
6 simp2 1136 . . . 4 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ ℂ)
7 fveq2 6907 . . . . . . . 8 (𝑔 = 𝐾 → (Base‘𝑔) = (Base‘𝐾))
87eqeq1d 2737 . . . . . . 7 (𝑔 = 𝐾 → ((Base‘𝑔) = ℂ ↔ (Base‘𝐾) = ℂ))
98, 1elrab2 3698 . . . . . 6 (𝐾𝐶 ↔ (𝐾 ∈ Abel ∧ (Base‘𝐾) = ℂ))
109simprbi 496 . . . . 5 (𝐾𝐶 → (Base‘𝐾) = ℂ)
11103ad2ant1 1132 . . . 4 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (Base‘𝐾) = ℂ)
126, 11eleqtrrd 2842 . . 3 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ (Base‘𝐾))
13 simp3 1137 . . . 4 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ)
1413, 11eleqtrrd 2842 . . 3 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ (Base‘𝐾))
15 eqid 2735 . . . 4 (Base‘𝐾) = (Base‘𝐾)
16 eqid 2735 . . . 4 (+g𝐾) = (+g𝐾)
1715, 16ablcom 19832 . . 3 ((𝐾 ∈ Abel ∧ 𝐴 ∈ (Base‘𝐾) ∧ 𝐵 ∈ (Base‘𝐾)) → (𝐴(+g𝐾)𝐵) = (𝐵(+g𝐾)𝐴))
185, 12, 14, 17syl3anc 1370 . 2 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴(+g𝐾)𝐵) = (𝐵(+g𝐾)𝐴))
19 toycom.2 . . 3 + = (+g𝐾)
2019oveqi 7444 . 2 (𝐴 + 𝐵) = (𝐴(+g𝐾)𝐵)
2119oveqi 7444 . 2 (𝐵 + 𝐴) = (𝐵(+g𝐾)𝐴)
2218, 20, 213eqtr4g 2800 1 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1537  wcel 2106  {crab 3433  cfv 6563  (class class class)co 7431  cc 11151  Basecbs 17245  +gcplusg 17298  Abelcabl 19814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-cmn 19815  df-abl 19816
This theorem is referenced by: (None)
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