Theorem toycom 39145

Description: Show the commutative law for an operation 𝑂 on a toy structure class 𝐶 of commutative 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	⊢ + = (+g‘𝐾)

Assertion

Ref	Expression
toycom	⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ 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

Step	Hyp	Ref	Expression
1		toycom.1	. . . . . 6 ⊢ 𝐶 = {𝑔 ∈ Abel ∣ (Base‘𝑔) = ℂ}
2		ssrab2 4029	. . . . . 6 ⊢ {𝑔 ∈ Abel ∣ (Base‘𝑔) = ℂ} ⊆ Abel
3	1, 2	eqsstri 3977	. . . . 5 ⊢ 𝐶 ⊆ Abel
4	3	sseli 3926	. . . 4 ⊢ (𝐾 ∈ 𝐶 → 𝐾 ∈ Abel)
5	4	3ad2ant1 1133	. . 3 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐾 ∈ Abel)
6		simp2 1137	. . . 4 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ ℂ)
7		fveq2 6831	. . . . . . . 8 ⊢ (𝑔 = 𝐾 → (Base‘𝑔) = (Base‘𝐾))
8	7	eqeq1d 2735	. . . . . . 7 ⊢ (𝑔 = 𝐾 → ((Base‘𝑔) = ℂ ↔ (Base‘𝐾) = ℂ))
9	8, 1	elrab2 3646	. . . . . 6 ⊢ (𝐾 ∈ 𝐶 ↔ (𝐾 ∈ Abel ∧ (Base‘𝐾) = ℂ))
10	9	simprbi 496	. . . . 5 ⊢ (𝐾 ∈ 𝐶 → (Base‘𝐾) = ℂ)
11	10	3ad2ant1 1133	. . . 4 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (Base‘𝐾) = ℂ)
12	6, 11	eleqtrrd 2836	. . 3 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ (Base‘𝐾))
13		simp3 1138	. . . 4 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ)
14	13, 11	eleqtrrd 2836	. . 3 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ (Base‘𝐾))
15		eqid 2733	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
16		eqid 2733	. . . 4 ⊢ (+g‘𝐾) = (+g‘𝐾)
17	15, 16	ablcom 19719	. . 3 ⊢ ((𝐾 ∈ Abel ∧ 𝐴 ∈ (Base‘𝐾) ∧ 𝐵 ∈ (Base‘𝐾)) → (𝐴(+g‘𝐾)𝐵) = (𝐵(+g‘𝐾)𝐴))
18	5, 12, 14, 17	syl3anc 1373	. 2 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴(+g‘𝐾)𝐵) = (𝐵(+g‘𝐾)𝐴))
19		toycom.2	. . 3 ⊢ + = (+g‘𝐾)
20	19	oveqi 7368	. 2 ⊢ (𝐴 + 𝐵) = (𝐴(+g‘𝐾)𝐵)
21	19	oveqi 7368	. 2 ⊢ (𝐵 + 𝐴) = (𝐵(+g‘𝐾)𝐴)
22	18, 20, 21	3eqtr4g 2793	1 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  {crab 3396  ‘cfv 6489  (class class class)co 7355  ℂcc 11015  Basecbs 17127  +_gcplusg 17168  Abelcabl 19701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-12 2182  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-iota 6445  df-fv 6497  df-ov 7358  df-cmn 19702  df-abl 19703

This theorem is referenced by: (None)