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Mirrors > Home > ILE Home > Th. List > 9t6e54 | GIF version |
Description: 9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
9t6e54 | ⊢ (9 · 6) = ;54 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 9nn0 8379 | . 2 ⊢ 9 ∈ ℕ0 | |
2 | 5nn0 8375 | . 2 ⊢ 5 ∈ ℕ0 | |
3 | df-6 8169 | . 2 ⊢ 6 = (5 + 1) | |
4 | 9t5e45 8682 | . 2 ⊢ (9 · 5) = ;45 | |
5 | 4nn0 8374 | . . 3 ⊢ 4 ∈ ℕ0 | |
6 | eqid 2082 | . . 3 ⊢ ;45 = ;45 | |
7 | 4p1e5 8235 | . . 3 ⊢ (4 + 1) = 5 | |
8 | 1 | nn0cni 8367 | . . . 4 ⊢ 9 ∈ ℂ |
9 | 2 | nn0cni 8367 | . . . 4 ⊢ 5 ∈ ℂ |
10 | 9p5e14 8647 | . . . 4 ⊢ (9 + 5) = ;14 | |
11 | 8, 9, 10 | addcomli 7320 | . . 3 ⊢ (5 + 9) = ;14 |
12 | 5, 2, 1, 6, 7, 5, 11 | decaddci 8618 | . 2 ⊢ (;45 + 9) = ;54 |
13 | 1, 2, 3, 4, 12 | 4t3lem 8654 | 1 ⊢ (9 · 6) = ;54 |
Colors of variables: wff set class |
Syntax hints: = wceq 1285 (class class class)co 5543 1c1 7044 · cmul 7048 4c4 8158 5c5 8159 6c6 8160 9c9 8163 ;cdc 8558 |
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 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-sep 3904 ax-pow 3956 ax-pr 3972 ax-setind 4288 ax-cnex 7129 ax-resscn 7130 ax-1cn 7131 ax-1re 7132 ax-icn 7133 ax-addcl 7134 ax-addrcl 7135 ax-mulcl 7136 ax-addcom 7138 ax-mulcom 7139 ax-addass 7140 ax-mulass 7141 ax-distr 7142 ax-i2m1 7143 ax-1rid 7145 ax-0id 7146 ax-rnegex 7147 ax-cnre 7149 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ne 2247 df-ral 2354 df-rex 2355 df-reu 2356 df-rab 2358 df-v 2604 df-sbc 2817 df-dif 2976 df-un 2978 df-in 2980 df-ss 2987 df-pw 3392 df-sn 3412 df-pr 3413 df-op 3415 df-uni 3610 df-int 3645 df-br 3794 df-opab 3848 df-id 4056 df-xp 4377 df-rel 4378 df-cnv 4379 df-co 4380 df-dm 4381 df-iota 4897 df-fun 4934 df-fv 4940 df-riota 5499 df-ov 5546 df-oprab 5547 df-mpt2 5548 df-sub 7348 df-inn 8107 df-2 8165 df-3 8166 df-4 8167 df-5 8168 df-6 8169 df-7 8170 df-8 8171 df-9 8172 df-n0 8356 df-dec 8559 |
This theorem is referenced by: 9t7e63 8684 |
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