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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 4056 | . . . . . 6 ⊢ {𝑔 ∈ Abel ∣ (Base‘𝑔) = ℂ} ⊆ Abel | |
3 | 1, 2 | eqsstri 4001 | . . . . 5 ⊢ 𝐶 ⊆ Abel |
4 | 3 | sseli 3963 | . . . 4 ⊢ (𝐾 ∈ 𝐶 → 𝐾 ∈ Abel) |
5 | 4 | 3ad2ant1 1129 | . . 3 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐾 ∈ Abel) |
6 | simp2 1133 | . . . 4 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ ℂ) | |
7 | fveq2 6665 | . . . . . . . 8 ⊢ (𝑔 = 𝐾 → (Base‘𝑔) = (Base‘𝐾)) | |
8 | 7 | eqeq1d 2823 | . . . . . . 7 ⊢ (𝑔 = 𝐾 → ((Base‘𝑔) = ℂ ↔ (Base‘𝐾) = ℂ)) |
9 | 8, 1 | elrab2 3683 | . . . . . 6 ⊢ (𝐾 ∈ 𝐶 ↔ (𝐾 ∈ Abel ∧ (Base‘𝐾) = ℂ)) |
10 | 9 | simprbi 499 | . . . . 5 ⊢ (𝐾 ∈ 𝐶 → (Base‘𝐾) = ℂ) |
11 | 10 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (Base‘𝐾) = ℂ) |
12 | 6, 11 | eleqtrrd 2916 | . . 3 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ (Base‘𝐾)) |
13 | simp3 1134 | . . . 4 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ) | |
14 | 13, 11 | eleqtrrd 2916 | . . 3 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ (Base‘𝐾)) |
15 | eqid 2821 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
16 | eqid 2821 | . . . 4 ⊢ (+g‘𝐾) = (+g‘𝐾) | |
17 | 15, 16 | ablcom 18918 | . . 3 ⊢ ((𝐾 ∈ Abel ∧ 𝐴 ∈ (Base‘𝐾) ∧ 𝐵 ∈ (Base‘𝐾)) → (𝐴(+g‘𝐾)𝐵) = (𝐵(+g‘𝐾)𝐴)) |
18 | 5, 12, 14, 17 | syl3anc 1367 | . 2 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴(+g‘𝐾)𝐵) = (𝐵(+g‘𝐾)𝐴)) |
19 | toycom.2 | . . 3 ⊢ + = (+g‘𝐾) | |
20 | 19 | oveqi 7163 | . 2 ⊢ (𝐴 + 𝐵) = (𝐴(+g‘𝐾)𝐵) |
21 | 19 | oveqi 7163 | . 2 ⊢ (𝐵 + 𝐴) = (𝐵(+g‘𝐾)𝐴) |
22 | 18, 20, 21 | 3eqtr4g 2881 | 1 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 {crab 3142 ‘cfv 6350 (class class class)co 7150 ℂcc 10529 Basecbs 16477 +gcplusg 16559 Abelcabl 18901 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-iota 6309 df-fv 6358 df-ov 7153 df-cmn 18902 df-abl 18903 |
This theorem is referenced by: (None) |
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