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Mirrors > Home > MPE Home > Th. List > xpundir | Structured version Visualization version GIF version |
Description: Distributive law for Cartesian product over union. Similar to Theorem 103 of [Suppes] p. 52. (Contributed by NM, 30-Sep-2002.) |
Ref | Expression |
---|---|
xpundir | ⊢ ((𝐴 ∪ 𝐵) × 𝐶) = ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xp 5044 | . 2 ⊢ ((𝐴 ∪ 𝐵) × 𝐶) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶)} | |
2 | df-xp 5044 | . . . 4 ⊢ (𝐴 × 𝐶) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)} | |
3 | df-xp 5044 | . . . 4 ⊢ (𝐵 × 𝐶) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)} | |
4 | 2, 3 | uneq12i 3727 | . . 3 ⊢ ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) = ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)} ∪ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)}) |
5 | elun 3715 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)) | |
6 | 5 | anbi1i 727 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐶)) |
7 | andir 908 | . . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∨ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) | |
8 | 6, 7 | bitri 263 | . . . . 5 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∨ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) |
9 | 8 | opabbii 4649 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶)} = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∨ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} |
10 | unopab 4660 | . . . 4 ⊢ ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)} ∪ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)}) = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∨ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} | |
11 | 9, 10 | eqtr4i 2635 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶)} = ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)} ∪ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)}) |
12 | 4, 11 | eqtr4i 2635 | . 2 ⊢ ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶)} |
13 | 1, 12 | eqtr4i 2635 | 1 ⊢ ((𝐴 ∪ 𝐵) × 𝐶) = ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∪ cun 3538 {copab 4642 × cxp 5036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-un 3545 df-opab 4644 df-xp 5044 |
This theorem is referenced by: xpun 5099 resundi 5330 xpfi 8116 cdaassen 8887 hashxplem 13080 ustund 21835 cnmpt2pc 22535 poimirlem3 32582 poimirlem4 32583 poimirlem6 32585 poimirlem7 32586 poimirlem16 32595 poimirlem19 32598 pwssplit4 36677 xpprsng 41903 |
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