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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 11348, see 1p2e3ALT 12298. (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 12217 | . . 3 ⊢ 2 = (1 + 1) | |
2 | 1 | oveq2i 7369 | . 2 ⊢ (1 + 2) = (1 + (1 + 1)) |
3 | ax-1cn 11110 | . . 3 ⊢ 1 ∈ ℂ | |
4 | 3, 3, 3 | addassi 11166 | . 2 ⊢ ((1 + 1) + 1) = (1 + (1 + 1)) |
5 | 1p1e2 12279 | . . . 4 ⊢ (1 + 1) = 2 | |
6 | 5 | oveq1i 7368 | . . 3 ⊢ ((1 + 1) + 1) = (2 + 1) |
7 | 2p1e3 12296 | . . 3 ⊢ (2 + 1) = 3 | |
8 | 6, 7 | eqtri 2765 | . 2 ⊢ ((1 + 1) + 1) = 3 |
9 | 2, 4, 8 | 3eqtr2i 2771 | 1 ⊢ (1 + 2) = 3 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 (class class class)co 7358 1c1 11053 + caddc 11055 2c2 12209 3c3 12210 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-1cn 11110 ax-addass 11117 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-iota 6449 df-fv 6505 df-ov 7361 df-2 12217 df-3 12218 |
This theorem is referenced by: fzo1to4tp 13661 binom3 14128 3lcm2e6woprm 16492 prmgaplem7 16930 2exp16 16964 prmlem1a 16980 23prm 16992 prmlem2 16993 83prm 16996 139prm 16997 163prm 16998 317prm 16999 631prm 17000 1259lem4 17007 1259prm 17009 2503lem2 17011 2503lem3 17012 4001lem2 17015 quart1lem 26208 log2ublem3 26301 log2ub 26302 pntibndlem2 26942 1kp2ke3k 29393 ex-ind-dvds 29408 fib4 33007 2np3bcnp1 40555 2xp3dxp2ge1d 40617 ex-decpmul 40809 sn-0ne2 40878 3cubeslem3r 41013 rabren3dioph 41141 fmtno4nprmfac193 45773 139prmALT 45795 127prm 45798 nnsum4primesodd 45995 nnsum4primesoddALTV 45996 ackval1012 46783 |
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