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Mirrors > Home > ILE Home > Th. List > 9t3e27 | GIF version |
Description: 9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
9t3e27 | ⊢ (9 · 3) = ;27 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 9nn0 8312 | . 2 ⊢ 9 ∈ ℕ0 | |
2 | 2nn0 8305 | . 2 ⊢ 2 ∈ ℕ0 | |
3 | df-3 8099 | . 2 ⊢ 3 = (2 + 1) | |
4 | 9t2e18 8598 | . 2 ⊢ (9 · 2) = ;18 | |
5 | 1nn0 8304 | . . 3 ⊢ 1 ∈ ℕ0 | |
6 | 8nn0 8311 | . . 3 ⊢ 8 ∈ ℕ0 | |
7 | eqid 2081 | . . 3 ⊢ ;18 = ;18 | |
8 | 1p1e2 8155 | . . 3 ⊢ (1 + 1) = 2 | |
9 | 7nn0 8310 | . . 3 ⊢ 7 ∈ ℕ0 | |
10 | 1 | nn0cni 8300 | . . . 4 ⊢ 9 ∈ ℂ |
11 | 6 | nn0cni 8300 | . . . 4 ⊢ 8 ∈ ℂ |
12 | 9p8e17 8569 | . . . 4 ⊢ (9 + 8) = ;17 | |
13 | 10, 11, 12 | addcomli 7253 | . . 3 ⊢ (8 + 9) = ;17 |
14 | 5, 6, 1, 7, 8, 9, 13 | decaddci 8537 | . 2 ⊢ (;18 + 9) = ;27 |
15 | 1, 2, 3, 4, 14 | 4t3lem 8573 | 1 ⊢ (9 · 3) = ;27 |
Colors of variables: wff set class |
Syntax hints: = wceq 1284 (class class class)co 5532 1c1 6982 · cmul 6986 2c2 8089 3c3 8090 7c7 8094 8c8 8095 9c9 8096 ;cdc 8477 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-sub 7281 df-inn 8040 df-2 8098 df-3 8099 df-4 8100 df-5 8101 df-6 8102 df-7 8103 df-8 8104 df-9 8105 df-n0 8289 df-dec 8478 |
This theorem is referenced by: 9t4e36 8600 |
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