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Theorem toycom 36262
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 4010 . . . . . 6 {𝑔 ∈ Abel ∣ (Base‘𝑔) = ℂ} ⊆ Abel
31, 2eqsstri 3952 . . . . 5 𝐶 ⊆ Abel
43sseli 3914 . . . 4 (𝐾𝐶𝐾 ∈ Abel)
543ad2ant1 1130 . . 3 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐾 ∈ Abel)
6 simp2 1134 . . . 4 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ ℂ)
7 fveq2 6649 . . . . . . . 8 (𝑔 = 𝐾 → (Base‘𝑔) = (Base‘𝐾))
87eqeq1d 2803 . . . . . . 7 (𝑔 = 𝐾 → ((Base‘𝑔) = ℂ ↔ (Base‘𝐾) = ℂ))
98, 1elrab2 3634 . . . . . 6 (𝐾𝐶 ↔ (𝐾 ∈ Abel ∧ (Base‘𝐾) = ℂ))
109simprbi 500 . . . . 5 (𝐾𝐶 → (Base‘𝐾) = ℂ)
11103ad2ant1 1130 . . . 4 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (Base‘𝐾) = ℂ)
126, 11eleqtrrd 2896 . . 3 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ (Base‘𝐾))
13 simp3 1135 . . . 4 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ)
1413, 11eleqtrrd 2896 . . 3 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ (Base‘𝐾))
15 eqid 2801 . . . 4 (Base‘𝐾) = (Base‘𝐾)
16 eqid 2801 . . . 4 (+g𝐾) = (+g𝐾)
1715, 16ablcom 18919 . . 3 ((𝐾 ∈ Abel ∧ 𝐴 ∈ (Base‘𝐾) ∧ 𝐵 ∈ (Base‘𝐾)) → (𝐴(+g𝐾)𝐵) = (𝐵(+g𝐾)𝐴))
185, 12, 14, 17syl3anc 1368 . 2 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴(+g𝐾)𝐵) = (𝐵(+g𝐾)𝐴))
19 toycom.2 . . 3 + = (+g𝐾)
2019oveqi 7152 . 2 (𝐴 + 𝐵) = (𝐴(+g𝐾)𝐵)
2119oveqi 7152 . 2 (𝐵 + 𝐴) = (𝐵(+g𝐾)𝐴)
2218, 20, 213eqtr4g 2861 1 ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2112  {crab 3113  cfv 6328  (class class class)co 7139  cc 10528  Basecbs 16478  +gcplusg 16560  Abelcabl 18902
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rab 3118  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-iota 6287  df-fv 6336  df-ov 7142  df-cmn 18903  df-abl 18904
This theorem is referenced by: (None)
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