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Theorem toycom 34578
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 3720 . . . . . 6 {𝑔 ∈ Abel ∣ (Base‘𝑔) = ℂ} ⊆ Abel
31, 2eqsstri 3668 . . . . 5 𝐶 ⊆ Abel
43sseli 3632 . . . 4 (𝐾𝐶𝐾 ∈ Abel)
543ad2ant1 1102 . . 3 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐾 ∈ Abel)
6 simp2 1082 . . . 4 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ ℂ)
7 fveq2 6229 . . . . . . . 8 (𝑔 = 𝐾 → (Base‘𝑔) = (Base‘𝐾))
87eqeq1d 2653 . . . . . . 7 (𝑔 = 𝐾 → ((Base‘𝑔) = ℂ ↔ (Base‘𝐾) = ℂ))
98, 1elrab2 3399 . . . . . 6 (𝐾𝐶 ↔ (𝐾 ∈ Abel ∧ (Base‘𝐾) = ℂ))
109simprbi 479 . . . . 5 (𝐾𝐶 → (Base‘𝐾) = ℂ)
11103ad2ant1 1102 . . . 4 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (Base‘𝐾) = ℂ)
126, 11eleqtrrd 2733 . . 3 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ (Base‘𝐾))
13 simp3 1083 . . . 4 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ)
1413, 11eleqtrrd 2733 . . 3 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ (Base‘𝐾))
15 eqid 2651 . . . 4 (Base‘𝐾) = (Base‘𝐾)
16 eqid 2651 . . . 4 (+g𝐾) = (+g𝐾)
1715, 16ablcom 18256 . . 3 ((𝐾 ∈ Abel ∧ 𝐴 ∈ (Base‘𝐾) ∧ 𝐵 ∈ (Base‘𝐾)) → (𝐴(+g𝐾)𝐵) = (𝐵(+g𝐾)𝐴))
185, 12, 14, 17syl3anc 1366 . 2 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴(+g𝐾)𝐵) = (𝐵(+g𝐾)𝐴))
19 toycom.2 . . 3 + = (+g𝐾)
2019oveqi 6703 . 2 (𝐴 + 𝐵) = (𝐴(+g𝐾)𝐵)
2119oveqi 6703 . 2 (𝐵 + 𝐴) = (𝐵(+g𝐾)𝐴)
2218, 20, 213eqtr4g 2710 1 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1054   = wceq 1523  wcel 2030  {crab 2945  cfv 5926  (class class class)co 6690  cc 9972  Basecbs 15904  +gcplusg 15988  Abelcabl 18240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-ov 6693  df-cmn 18241  df-abl 18242
This theorem is referenced by: (None)
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