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Mirrors > Home > MPE Home > Th. List > 1p2e3 | Structured version Visualization version GIF version |
Description: 1 + 2 = 3. For a shorter proof using addcomli 10820, see 1p2e3ALT 11769. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 12-Dec-2022.) |
Ref | Expression |
---|---|
1p2e3 | ⊢ (1 + 2) = 3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2 11688 | . . 3 ⊢ 2 = (1 + 1) | |
2 | 1 | oveq2i 7156 | . 2 ⊢ (1 + 2) = (1 + (1 + 1)) |
3 | ax-1cn 10583 | . . 3 ⊢ 1 ∈ ℂ | |
4 | 3, 3, 3 | addassi 10639 | . 2 ⊢ ((1 + 1) + 1) = (1 + (1 + 1)) |
5 | 1p1e2 11750 | . . . 4 ⊢ (1 + 1) = 2 | |
6 | 5 | oveq1i 7155 | . . 3 ⊢ ((1 + 1) + 1) = (2 + 1) |
7 | 2p1e3 11767 | . . 3 ⊢ (2 + 1) = 3 | |
8 | 6, 7 | eqtri 2841 | . 2 ⊢ ((1 + 1) + 1) = 3 |
9 | 2, 4, 8 | 3eqtr2i 2847 | 1 ⊢ (1 + 2) = 3 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 (class class class)co 7145 1c1 10526 + caddc 10528 2c2 11680 3c3 11681 |
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-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 |
This theorem is referenced by: fzo1to4tp 13113 binom3 13573 3lcm2e6woprm 15947 prmgaplem7 16381 2exp16 16412 prmlem1a 16428 23prm 16440 prmlem2 16441 83prm 16444 139prm 16445 163prm 16446 317prm 16447 631prm 16448 1259lem4 16455 1259prm 16457 2503lem2 16459 2503lem3 16460 4001lem2 16463 quart1lem 25360 log2ublem3 25453 log2ub 25454 pntibndlem2 26094 1kp2ke3k 28152 ex-ind-dvds 28167 fib4 31561 ex-decpmul 39056 sn-0ne2 39114 3cubeslem3r 39162 rabren3dioph 39290 fmtno4nprmfac193 43613 139prmALT 43636 127prm 43640 nnsum4primesodd 43838 nnsum4primesoddALTV 43839 |
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