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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 5554 | . 2 ⊢ ((𝐴 ∪ 𝐵) × 𝐶) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶)} | |
2 | df-xp 5554 | . . . 4 ⊢ (𝐴 × 𝐶) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)} | |
3 | df-xp 5554 | . . . 4 ⊢ (𝐵 × 𝐶) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)} | |
4 | 2, 3 | uneq12i 4130 | . . 3 ⊢ ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) = ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)} ∪ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)}) |
5 | elun 4118 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)) | |
6 | 5 | anbi1i 625 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐶)) |
7 | andir 1005 | . . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∨ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) | |
8 | 6, 7 | bitri 277 | . . . . 5 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∨ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) |
9 | 8 | opabbii 5126 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶)} = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∨ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} |
10 | unopab 5138 | . . . 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 2846 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶)} = ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)} ∪ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)}) |
12 | 4, 11 | eqtr4i 2846 | . 2 ⊢ ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶)} |
13 | 1, 12 | eqtr4i 2846 | 1 ⊢ ((𝐴 ∪ 𝐵) × 𝐶) = ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∨ wo 843 = wceq 1536 ∈ wcel 2113 ∪ cun 3927 {copab 5121 × cxp 5546 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-v 3493 df-un 3934 df-opab 5122 df-xp 5554 |
This theorem is referenced by: xpun 5618 resundi 5860 xpprsng 6895 xpfi 8782 xp2dju 9595 alephadd 9992 hashxplem 13791 ustund 22825 cnmpopc 23527 poimirlem3 34930 poimirlem4 34931 poimirlem6 34933 poimirlem7 34934 poimirlem16 34943 poimirlem19 34946 pwssplit4 39765 |
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