Theorem toycom 39419

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 4020	. . . . . 6 ⊢ {𝑔 ∈ Abel ∣ (Base‘𝑔) = ℂ} ⊆ Abel
3	1, 2	eqsstri 3968	. . . . 5 ⊢ 𝐶 ⊆ Abel
4	3	sseli 3917	. . . 4 ⊢ (𝐾 ∈ 𝐶 → 𝐾 ∈ Abel)
5	4	3ad2ant1 1134	. . 3 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐾 ∈ Abel)
6		simp2 1138	. . . 4 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ ℂ)
7		fveq2 6840	. . . . . . . 8 ⊢ (𝑔 = 𝐾 → (Base‘𝑔) = (Base‘𝐾))
8	7	eqeq1d 2738	. . . . . . 7 ⊢ (𝑔 = 𝐾 → ((Base‘𝑔) = ℂ ↔ (Base‘𝐾) = ℂ))
9	8, 1	elrab2 3637	. . . . . 6 ⊢ (𝐾 ∈ 𝐶 ↔ (𝐾 ∈ Abel ∧ (Base‘𝐾) = ℂ))
10	9	simprbi 497	. . . . 5 ⊢ (𝐾 ∈ 𝐶 → (Base‘𝐾) = ℂ)
11	10	3ad2ant1 1134	. . . 4 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (Base‘𝐾) = ℂ)
12	6, 11	eleqtrrd 2839	. . 3 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ (Base‘𝐾))
13		simp3 1139	. . . 4 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ)
14	13, 11	eleqtrrd 2839	. . 3 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ (Base‘𝐾))
15		eqid 2736	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
16		eqid 2736	. . . 4 ⊢ (+g‘𝐾) = (+g‘𝐾)
17	15, 16	ablcom 19774	. . 3 ⊢ ((𝐾 ∈ Abel ∧ 𝐴 ∈ (Base‘𝐾) ∧ 𝐵 ∈ (Base‘𝐾)) → (𝐴(+g‘𝐾)𝐵) = (𝐵(+g‘𝐾)𝐴))
18	5, 12, 14, 17	syl3anc 1374	. 2 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴(+g‘𝐾)𝐵) = (𝐵(+g‘𝐾)𝐴))
19		toycom.2	. . 3 ⊢ + = (+g‘𝐾)
20	19	oveqi 7380	. 2 ⊢ (𝐴 + 𝐵) = (𝐴(+g‘𝐾)𝐵)
21	19	oveqi 7380	. 2 ⊢ (𝐵 + 𝐴) = (𝐵(+g‘𝐾)𝐴)
22	18, 20, 21	3eqtr4g 2796	1 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {crab 3389  ‘cfv 6498  (class class class)co 7367  ℂcc 11036  Basecbs 17179  +_gcplusg 17220  Abelcabl 19756

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-12 2185  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-ov 7370  df-cmn 19757  df-abl 19758

This theorem is referenced by: (None)