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Mirrors > Home > ILE Home > Th. List > 9t7e63 | GIF version |
Description: 9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
9t7e63 | ⊢ (9 · 7) = ;63 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 9nn0 8368 | . 2 ⊢ 9 ∈ ℕ0 | |
2 | 6nn0 8365 | . 2 ⊢ 6 ∈ ℕ0 | |
3 | df-7 8159 | . 2 ⊢ 7 = (6 + 1) | |
4 | 9t6e54 8672 | . 2 ⊢ (9 · 6) = ;54 | |
5 | 5nn0 8364 | . . 3 ⊢ 5 ∈ ℕ0 | |
6 | 4nn0 8363 | . . 3 ⊢ 4 ∈ ℕ0 | |
7 | eqid 2082 | . . 3 ⊢ ;54 = ;54 | |
8 | 5p1e6 8225 | . . 3 ⊢ (5 + 1) = 6 | |
9 | 3nn0 8362 | . . 3 ⊢ 3 ∈ ℕ0 | |
10 | 1 | nn0cni 8356 | . . . 4 ⊢ 9 ∈ ℂ |
11 | 6 | nn0cni 8356 | . . . 4 ⊢ 4 ∈ ℂ |
12 | 9p4e13 8635 | . . . 4 ⊢ (9 + 4) = ;13 | |
13 | 10, 11, 12 | addcomli 7309 | . . 3 ⊢ (4 + 9) = ;13 |
14 | 5, 6, 1, 7, 8, 9, 13 | decaddci 8607 | . 2 ⊢ (;54 + 9) = ;63 |
15 | 1, 2, 3, 4, 14 | 4t3lem 8643 | 1 ⊢ (9 · 7) = ;63 |
Colors of variables: wff set class |
Syntax hints: = wceq 1285 (class class class)co 5537 1c1 7033 · cmul 7037 3c3 8146 4c4 8147 5c5 8148 6c6 8149 7c7 8150 9c9 8152 ;cdc 8547 |
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 3898 ax-pow 3950 ax-pr 3966 ax-setind 4282 ax-cnex 7118 ax-resscn 7119 ax-1cn 7120 ax-1re 7121 ax-icn 7122 ax-addcl 7123 ax-addrcl 7124 ax-mulcl 7125 ax-addcom 7127 ax-mulcom 7128 ax-addass 7129 ax-mulass 7130 ax-distr 7131 ax-i2m1 7132 ax-1rid 7134 ax-0id 7135 ax-rnegex 7136 ax-cnre 7138 |
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 3386 df-sn 3406 df-pr 3407 df-op 3409 df-uni 3604 df-int 3639 df-br 3788 df-opab 3842 df-id 4050 df-xp 4371 df-rel 4372 df-cnv 4373 df-co 4374 df-dm 4375 df-iota 4891 df-fun 4928 df-fv 4934 df-riota 5493 df-ov 5540 df-oprab 5541 df-mpt2 5542 df-sub 7337 df-inn 8096 df-2 8154 df-3 8155 df-4 8156 df-5 8157 df-6 8158 df-7 8159 df-8 8160 df-9 8161 df-n0 8345 df-dec 8548 |
This theorem is referenced by: 9t8e72 8674 |
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