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Mirrors > Home > ILE Home > Th. List > inxp | GIF version |
Description: The intersection of two cross products. Exercise 9 of [TakeutiZaring] p. 25. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
inxp | ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inopab 4794 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)} ∩ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}) = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷))} | |
2 | an4 586 | . . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷))) | |
3 | elin 3342 | . . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶)) | |
4 | elin 3342 | . . . . . 6 ⊢ (𝑦 ∈ (𝐵 ∩ 𝐷) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷)) | |
5 | 3, 4 | anbi12i 460 | . . . . 5 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐶) ∧ 𝑦 ∈ (𝐵 ∩ 𝐷)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷))) |
6 | 2, 5 | bitr4i 187 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) ↔ (𝑥 ∈ (𝐴 ∩ 𝐶) ∧ 𝑦 ∈ (𝐵 ∩ 𝐷))) |
7 | 6 | opabbii 4096 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷))} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (𝐴 ∩ 𝐶) ∧ 𝑦 ∈ (𝐵 ∩ 𝐷))} |
8 | 1, 7 | eqtri 2214 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)} ∩ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (𝐴 ∩ 𝐶) ∧ 𝑦 ∈ (𝐵 ∩ 𝐷))} |
9 | df-xp 4665 | . . 3 ⊢ (𝐴 × 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)} | |
10 | df-xp 4665 | . . 3 ⊢ (𝐶 × 𝐷) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} | |
11 | 9, 10 | ineq12i 3358 | . 2 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) = ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)} ∩ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}) |
12 | df-xp 4665 | . 2 ⊢ ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷)) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (𝐴 ∩ 𝐶) ∧ 𝑦 ∈ (𝐵 ∩ 𝐷))} | |
13 | 8, 11, 12 | 3eqtr4i 2224 | 1 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1364 ∈ wcel 2164 ∩ cin 3152 {copab 4089 × cxp 4657 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-opab 4091 df-xp 4665 df-rel 4666 |
This theorem is referenced by: xpindi 4797 xpindir 4798 dmxpin 4884 xpssres 4977 xpdisj1 5090 xpdisj2 5091 imainrect 5111 xpima1 5112 xpima2m 5113 hashxp 10897 txbas 14426 txrest 14444 metreslem 14548 |
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