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Mirrors > Home > ILE Home > Th. List > 7t4e28 | GIF version |
Description: 7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
7t4e28 | ⊢ (7 · 4) = ;28 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 7nn0 8377 | . 2 ⊢ 7 ∈ ℕ0 | |
2 | 3nn0 8373 | . 2 ⊢ 3 ∈ ℕ0 | |
3 | df-4 8167 | . 2 ⊢ 4 = (3 + 1) | |
4 | 7t3e21 8667 | . 2 ⊢ (7 · 3) = ;21 | |
5 | 2nn0 8372 | . . 3 ⊢ 2 ∈ ℕ0 | |
6 | 1nn0 8371 | . . 3 ⊢ 1 ∈ ℕ0 | |
7 | eqid 2082 | . . 3 ⊢ ;21 = ;21 | |
8 | 7cn 8190 | . . . 4 ⊢ 7 ∈ ℂ | |
9 | ax-1cn 7131 | . . . 4 ⊢ 1 ∈ ℂ | |
10 | 7p1e8 8238 | . . . 4 ⊢ (7 + 1) = 8 | |
11 | 8, 9, 10 | addcomli 7320 | . . 3 ⊢ (1 + 7) = 8 |
12 | 5, 6, 1, 7, 11 | decaddi 8617 | . 2 ⊢ (;21 + 7) = ;28 |
13 | 1, 2, 3, 4, 12 | 4t3lem 8654 | 1 ⊢ (7 · 4) = ;28 |
Colors of variables: wff set class |
Syntax hints: = wceq 1285 (class class class)co 5543 1c1 7044 · cmul 7048 2c2 8156 3c3 8157 4c4 8158 7c7 8161 8c8 8162 ;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: 7t5e35 8669 |
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