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Mathbox for Norm Megill |
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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 3720 | . . . . . 6 ⊢ {𝑔 ∈ Abel ∣ (Base‘𝑔) = ℂ} ⊆ Abel | |
3 | 1, 2 | eqsstri 3668 | . . . . 5 ⊢ 𝐶 ⊆ Abel |
4 | 3 | sseli 3632 | . . . 4 ⊢ (𝐾 ∈ 𝐶 → 𝐾 ∈ Abel) |
5 | 4 | 3ad2ant1 1102 | . . 3 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐾 ∈ Abel) |
6 | simp2 1082 | . . . 4 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ ℂ) | |
7 | fveq2 6229 | . . . . . . . 8 ⊢ (𝑔 = 𝐾 → (Base‘𝑔) = (Base‘𝐾)) | |
8 | 7 | eqeq1d 2653 | . . . . . . 7 ⊢ (𝑔 = 𝐾 → ((Base‘𝑔) = ℂ ↔ (Base‘𝐾) = ℂ)) |
9 | 8, 1 | elrab2 3399 | . . . . . 6 ⊢ (𝐾 ∈ 𝐶 ↔ (𝐾 ∈ Abel ∧ (Base‘𝐾) = ℂ)) |
10 | 9 | simprbi 479 | . . . . 5 ⊢ (𝐾 ∈ 𝐶 → (Base‘𝐾) = ℂ) |
11 | 10 | 3ad2ant1 1102 | . . . 4 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (Base‘𝐾) = ℂ) |
12 | 6, 11 | eleqtrrd 2733 | . . 3 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ (Base‘𝐾)) |
13 | simp3 1083 | . . . 4 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ) | |
14 | 13, 11 | eleqtrrd 2733 | . . 3 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ (Base‘𝐾)) |
15 | eqid 2651 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
16 | eqid 2651 | . . . 4 ⊢ (+g‘𝐾) = (+g‘𝐾) | |
17 | 15, 16 | ablcom 18256 | . . 3 ⊢ ((𝐾 ∈ Abel ∧ 𝐴 ∈ (Base‘𝐾) ∧ 𝐵 ∈ (Base‘𝐾)) → (𝐴(+g‘𝐾)𝐵) = (𝐵(+g‘𝐾)𝐴)) |
18 | 5, 12, 14, 17 | syl3anc 1366 | . 2 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴(+g‘𝐾)𝐵) = (𝐵(+g‘𝐾)𝐴)) |
19 | toycom.2 | . . 3 ⊢ + = (+g‘𝐾) | |
20 | 19 | oveqi 6703 | . 2 ⊢ (𝐴 + 𝐵) = (𝐴(+g‘𝐾)𝐵) |
21 | 19 | oveqi 6703 | . 2 ⊢ (𝐵 + 𝐴) = (𝐵(+g‘𝐾)𝐴) |
22 | 18, 20, 21 | 3eqtr4g 2710 | 1 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 {crab 2945 ‘cfv 5926 (class class class)co 6690 ℂcc 9972 Basecbs 15904 +gcplusg 15988 Abelcabl 18240 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-iota 5889 df-fv 5934 df-ov 6693 df-cmn 18241 df-abl 18242 |
This theorem is referenced by: (None) |
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