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Mirrors > Home > MPE Home > Th. List > 2p2e4 | Structured version Visualization version GIF version |
Description: Two plus two equals four. For more information, see "2+2=4 Trivia" on the Metamath Proof Explorer Home Page: mmset.html#trivia. This proof is simple, but it depends on many other proof steps because 2 and 4 are complex numbers and thus it depends on our construction of complex numbers. The proof o2p2e4 8155 is similar but proves 2 + 2 = 4 using ordinal natural numbers (finite integers starting at 0), so that proof depends on fewer intermediate steps. (Contributed by NM, 27-May-1999.) |
Ref | Expression |
---|---|
2p2e4 | ⊢ (2 + 2) = 4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2 11688 | . . 3 ⊢ 2 = (1 + 1) | |
2 | 1 | oveq2i 7156 | . 2 ⊢ (2 + 2) = (2 + (1 + 1)) |
3 | df-4 11690 | . . 3 ⊢ 4 = (3 + 1) | |
4 | df-3 11689 | . . . 4 ⊢ 3 = (2 + 1) | |
5 | 4 | oveq1i 7155 | . . 3 ⊢ (3 + 1) = ((2 + 1) + 1) |
6 | 2cn 11700 | . . . 4 ⊢ 2 ∈ ℂ | |
7 | ax-1cn 10583 | . . . 4 ⊢ 1 ∈ ℂ | |
8 | 6, 7, 7 | addassi 10639 | . . 3 ⊢ ((2 + 1) + 1) = (2 + (1 + 1)) |
9 | 3, 5, 8 | 3eqtri 2845 | . 2 ⊢ 4 = (2 + (1 + 1)) |
10 | 2, 9 | eqtr4i 2844 | 1 ⊢ (2 + 2) = 4 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 (class class class)co 7145 1c1 10526 + caddc 10528 2c2 11680 3c3 11681 4c4 11682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-1cn 10583 ax-addcl 10585 ax-addass 10590 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7148 df-2 11688 df-3 11689 df-4 11690 |
This theorem is referenced by: 2t2e4 11789 i4 13555 4bc2eq6 13677 bpoly4 15401 fsumcube 15402 ef01bndlem 15525 6gcd4e2 15874 pythagtriplem1 16141 prmlem2 16441 43prm 16443 1259lem4 16455 2503lem1 16458 2503lem2 16459 2503lem3 16460 4001lem1 16462 4001lem4 16465 cphipval2 23771 quart1lem 25360 log2ub 25454 hgt750lem2 31822 3cubeslem3l 39161 3cubeslem3r 39162 wallispi2lem1 42233 stirlinglem8 42243 sqwvfourb 42391 fmtnorec4 43588 m11nprm 43643 3exp4mod41 43658 gbowgt5 43804 gbpart7 43809 sbgoldbaltlem1 43821 sbgoldbalt 43823 sgoldbeven3prm 43825 mogoldbb 43827 nnsum3primes4 43830 2t6m3t4e0 44324 2p2ne5 44827 |
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