Theorem toycom 37832

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	⊢ + = (+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 4077	. . . . . 6 ⊢ {𝑔 ∈ Abel ∣ (Base‘𝑔) = ℂ} ⊆ Abel
3	1, 2	eqsstri 4016	. . . . 5 ⊢ 𝐶 ⊆ Abel
4	3	sseli 3978	. . . 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 6889	. . . . . . . 8 ⊢ (𝑔 = 𝐾 → (Base‘𝑔) = (Base‘𝐾))
8	7	eqeq1d 2735	. . . . . . 7 ⊢ (𝑔 = 𝐾 → ((Base‘𝑔) = ℂ ↔ (Base‘𝐾) = ℂ))
9	8, 1	elrab2 3686	. . . . . 6 ⊢ (𝐾 ∈ 𝐶 ↔ (𝐾 ∈ Abel ∧ (Base‘𝐾) = ℂ))
10	9	simprbi 498	. . . . 5 ⊢ (𝐾 ∈ 𝐶 → (Base‘𝐾) = ℂ)
11	10	3ad2ant1 1134	. . . 4 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (Base‘𝐾) = ℂ)
12	6, 11	eleqtrrd 2837	. . 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 2837	. . 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 19662	. . 3 ⊢ ((𝐾 ∈ Abel ∧ 𝐴 ∈ (Base‘𝐾) ∧ 𝐵 ∈ (Base‘𝐾)) → (𝐴(+g‘𝐾)𝐵) = (𝐵(+g‘𝐾)𝐴))
18	5, 12, 14, 17	syl3anc 1372	. 2 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴(+g‘𝐾)𝐵) = (𝐵(+g‘𝐾)𝐴))
19		toycom.2	. . 3 ⊢ + = (+g‘𝐾)
20	19	oveqi 7419	. 2 ⊢ (𝐴 + 𝐵) = (𝐴(+g‘𝐾)𝐵)
21	19	oveqi 7419	. 2 ⊢ (𝐵 + 𝐴) = (𝐵(+g‘𝐾)𝐴)
22	18, 20, 21	3eqtr4g 2798	1 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Syntax hints:   → wi 4   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {crab 3433  ‘cfv 6541  (class class class)co 7406  ℂcc 11105  Basecbs 17141  +_gcplusg 17194  Abelcabl 19644

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6493  df-fv 6549  df-ov 7409  df-cmn 19645  df-abl 19646

This theorem is referenced by: (None)