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Mirrors > Home > MPE Home > Th. List > Mathboxes > xpprsng | Structured version Visualization version GIF version |
Description: The Cartesian product of an unordered pair and a singleton. (Contributed by AV, 20-May-2019.) |
Ref | Expression |
---|---|
xpprsng | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑈) → ({𝐴, 𝐵} × {𝐶}) = {〈𝐴, 𝐶〉, 〈𝐵, 𝐶〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 4128 | . . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
2 | 1 | xpeq1i 5059 | . 2 ⊢ ({𝐴, 𝐵} × {𝐶}) = (({𝐴} ∪ {𝐵}) × {𝐶}) |
3 | xpsng 6312 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑈) → ({𝐴} × {𝐶}) = {〈𝐴, 𝐶〉}) | |
4 | 3 | 3adant2 1073 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑈) → ({𝐴} × {𝐶}) = {〈𝐴, 𝐶〉}) |
5 | xpsng 6312 | . . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑈) → ({𝐵} × {𝐶}) = {〈𝐵, 𝐶〉}) | |
6 | 5 | 3adant1 1072 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑈) → ({𝐵} × {𝐶}) = {〈𝐵, 𝐶〉}) |
7 | 4, 6 | uneq12d 3730 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑈) → (({𝐴} × {𝐶}) ∪ ({𝐵} × {𝐶})) = ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐶〉})) |
8 | xpundir 5095 | . . 3 ⊢ (({𝐴} ∪ {𝐵}) × {𝐶}) = (({𝐴} × {𝐶}) ∪ ({𝐵} × {𝐶})) | |
9 | df-pr 4128 | . . 3 ⊢ {〈𝐴, 𝐶〉, 〈𝐵, 𝐶〉} = ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐶〉}) | |
10 | 7, 8, 9 | 3eqtr4g 2669 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑈) → (({𝐴} ∪ {𝐵}) × {𝐶}) = {〈𝐴, 𝐶〉, 〈𝐵, 𝐶〉}) |
11 | 2, 10 | syl5eq 2656 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑈) → ({𝐴, 𝐵} × {𝐶}) = {〈𝐴, 𝐶〉, 〈𝐵, 𝐶〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∪ cun 3538 {csn 4125 {cpr 4127 〈cop 4131 × 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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 |
This theorem is referenced by: zlmodzxz0 41927 |
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