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Mirrors > Home > ILE Home > Th. List > 2p2e4 | GIF version |
Description: Two plus two equals four. For more information, see "2+2=4 Trivia" on the Metamath Proof Explorer Home Page: https://us.metamath.org/mpeuni/mmset.html#trivia. (Contributed by NM, 27-May-1999.) |
Ref | Expression |
---|---|
2p2e4 | ⊢ (2 + 2) = 4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2 8779 | . . 3 ⊢ 2 = (1 + 1) | |
2 | 1 | oveq2i 5785 | . 2 ⊢ (2 + 2) = (2 + (1 + 1)) |
3 | df-4 8781 | . . 3 ⊢ 4 = (3 + 1) | |
4 | df-3 8780 | . . . 4 ⊢ 3 = (2 + 1) | |
5 | 4 | oveq1i 5784 | . . 3 ⊢ (3 + 1) = ((2 + 1) + 1) |
6 | 2cn 8791 | . . . 4 ⊢ 2 ∈ ℂ | |
7 | ax-1cn 7713 | . . . 4 ⊢ 1 ∈ ℂ | |
8 | 6, 7, 7 | addassi 7774 | . . 3 ⊢ ((2 + 1) + 1) = (2 + (1 + 1)) |
9 | 3, 5, 8 | 3eqtri 2164 | . 2 ⊢ 4 = (2 + (1 + 1)) |
10 | 2, 9 | eqtr4i 2163 | 1 ⊢ (2 + 2) = 4 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 (class class class)co 5774 1c1 7621 + caddc 7623 2c2 8771 3c3 8772 4c4 8773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-addrcl 7717 ax-addass 7722 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-iota 5088 df-fv 5131 df-ov 5777 df-2 8779 df-3 8780 df-4 8781 |
This theorem is referenced by: 2t2e4 8874 i4 10395 4bc2eq6 10520 resqrexlemover 10782 resqrexlemcalc1 10786 ef01bndlem 11463 6gcd4e2 11683 |
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