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Mirrors > Home > MPE Home > Th. List > Mathboxes > toycom | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
toycom.1 | ⊢ 𝐶 = {𝑔 ∈ Abel ∣ (Base‘𝑔) = ℂ} |
toycom.2 | ⊢ + = (+g‘𝐾) |
Ref | Expression |
---|---|
toycom | ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
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)) |
Colors of variables: wff setvar class |
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) |
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