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Mirrors > Home > ILE Home > Th. List > 9t5e45 | GIF version |
Description: 9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
9t5e45 | ⊢ (9 · 5) = ;45 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 9nn0 9119 | . 2 ⊢ 9 ∈ ℕ0 | |
2 | 4nn0 9114 | . 2 ⊢ 4 ∈ ℕ0 | |
3 | df-5 8900 | . 2 ⊢ 5 = (4 + 1) | |
4 | 9t4e36 9423 | . 2 ⊢ (9 · 4) = ;36 | |
5 | 3nn0 9113 | . . 3 ⊢ 3 ∈ ℕ0 | |
6 | 6nn0 9116 | . . 3 ⊢ 6 ∈ ℕ0 | |
7 | eqid 2157 | . . 3 ⊢ ;36 = ;36 | |
8 | 3p1e4 8973 | . . 3 ⊢ (3 + 1) = 4 | |
9 | 5nn0 9115 | . . 3 ⊢ 5 ∈ ℕ0 | |
10 | 1 | nn0cni 9107 | . . . 4 ⊢ 9 ∈ ℂ |
11 | 6 | nn0cni 9107 | . . . 4 ⊢ 6 ∈ ℂ |
12 | 9p6e15 9390 | . . . 4 ⊢ (9 + 6) = ;15 | |
13 | 10, 11, 12 | addcomli 8024 | . . 3 ⊢ (6 + 9) = ;15 |
14 | 5, 6, 1, 7, 8, 9, 13 | decaddci 9360 | . 2 ⊢ (;36 + 9) = ;45 |
15 | 1, 2, 3, 4, 14 | 4t3lem 9396 | 1 ⊢ (9 · 5) = ;45 |
Colors of variables: wff set class |
Syntax hints: = wceq 1335 (class class class)co 5826 1c1 7735 · cmul 7739 3c3 8890 4c4 8891 5c5 8892 6c6 8893 9c9 8896 ;cdc 9300 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 ax-setind 4498 ax-cnex 7825 ax-resscn 7826 ax-1cn 7827 ax-1re 7828 ax-icn 7829 ax-addcl 7830 ax-addrcl 7831 ax-mulcl 7832 ax-addcom 7834 ax-mulcom 7835 ax-addass 7836 ax-mulass 7837 ax-distr 7838 ax-i2m1 7839 ax-1rid 7841 ax-0id 7842 ax-rnegex 7843 ax-cnre 7845 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-br 3968 df-opab 4028 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-iota 5137 df-fun 5174 df-fv 5180 df-riota 5782 df-ov 5829 df-oprab 5830 df-mpo 5831 df-sub 8052 df-inn 8839 df-2 8897 df-3 8898 df-4 8899 df-5 8900 df-6 8901 df-7 8902 df-8 8903 df-9 8904 df-n0 9096 df-dec 9301 |
This theorem is referenced by: 9t6e54 9425 |
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