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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 commutative 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 4080 | . . . . . 6 ⊢ {𝑔 ∈ Abel ∣ (Base‘𝑔) = ℂ} ⊆ Abel | |
| 3 | 1, 2 | eqsstri 4030 | . . . . 5 ⊢ 𝐶 ⊆ Abel |
| 4 | 3 | sseli 3979 | . . . 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 6906 | . . . . . . . 8 ⊢ (𝑔 = 𝐾 → (Base‘𝑔) = (Base‘𝐾)) | |
| 8 | 7 | eqeq1d 2739 | . . . . . . 7 ⊢ (𝑔 = 𝐾 → ((Base‘𝑔) = ℂ ↔ (Base‘𝐾) = ℂ)) |
| 9 | 8, 1 | elrab2 3695 | . . . . . 6 ⊢ (𝐾 ∈ 𝐶 ↔ (𝐾 ∈ Abel ∧ (Base‘𝐾) = ℂ)) |
| 10 | 9 | simprbi 496 | . . . . 5 ⊢ (𝐾 ∈ 𝐶 → (Base‘𝐾) = ℂ) |
| 11 | 10 | 3ad2ant1 1134 | . . . 4 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (Base‘𝐾) = ℂ) |
| 12 | 6, 11 | eleqtrrd 2844 | . . 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 2844 | . . 3 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ (Base‘𝐾)) |
| 15 | eqid 2737 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 16 | eqid 2737 | . . . 4 ⊢ (+g‘𝐾) = (+g‘𝐾) | |
| 17 | 15, 16 | ablcom 19817 | . . 3 ⊢ ((𝐾 ∈ Abel ∧ 𝐴 ∈ (Base‘𝐾) ∧ 𝐵 ∈ (Base‘𝐾)) → (𝐴(+g‘𝐾)𝐵) = (𝐵(+g‘𝐾)𝐴)) |
| 18 | 5, 12, 14, 17 | syl3anc 1373 | . 2 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴(+g‘𝐾)𝐵) = (𝐵(+g‘𝐾)𝐴)) |
| 19 | toycom.2 | . . 3 ⊢ + = (+g‘𝐾) | |
| 20 | 19 | oveqi 7444 | . 2 ⊢ (𝐴 + 𝐵) = (𝐴(+g‘𝐾)𝐵) |
| 21 | 19 | oveqi 7444 | . 2 ⊢ (𝐵 + 𝐴) = (𝐵(+g‘𝐾)𝐴) |
| 22 | 18, 20, 21 | 3eqtr4g 2802 | 1 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 {crab 3436 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 Basecbs 17247 +gcplusg 17297 Abelcabl 19799 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-cmn 19800 df-abl 19801 |
| This theorem is referenced by: (None) |
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