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Mirrors > Home > ILE Home > Th. List > 9t8e72 | GIF version |
Description: 9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
9t8e72 | ⊢ (9 · 8) = ;72 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 9nn0 8994 | . 2 ⊢ 9 ∈ ℕ0 | |
2 | 7nn0 8992 | . 2 ⊢ 7 ∈ ℕ0 | |
3 | df-8 8778 | . 2 ⊢ 8 = (7 + 1) | |
4 | 9t7e63 9301 | . 2 ⊢ (9 · 7) = ;63 | |
5 | 6nn0 8991 | . . 3 ⊢ 6 ∈ ℕ0 | |
6 | 3nn0 8988 | . . 3 ⊢ 3 ∈ ℕ0 | |
7 | eqid 2137 | . . 3 ⊢ ;63 = ;63 | |
8 | 6p1e7 8851 | . . 3 ⊢ (6 + 1) = 7 | |
9 | 2nn0 8987 | . . 3 ⊢ 2 ∈ ℕ0 | |
10 | 9cn 8801 | . . . 4 ⊢ 9 ∈ ℂ | |
11 | 3cn 8788 | . . . 4 ⊢ 3 ∈ ℂ | |
12 | 9p3e12 9262 | . . . 4 ⊢ (9 + 3) = ;12 | |
13 | 10, 11, 12 | addcomli 7900 | . . 3 ⊢ (3 + 9) = ;12 |
14 | 5, 6, 1, 7, 8, 9, 13 | decaddci 9235 | . 2 ⊢ (;63 + 9) = ;72 |
15 | 1, 2, 3, 4, 14 | 4t3lem 9271 | 1 ⊢ (9 · 8) = ;72 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 (class class class)co 5767 1c1 7614 · cmul 7618 2c2 8764 3c3 8765 6c6 8768 7c7 8769 8c8 8770 9c9 8771 ;cdc 9175 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-sub 7928 df-inn 8714 df-2 8772 df-3 8773 df-4 8774 df-5 8775 df-6 8776 df-7 8777 df-8 8778 df-9 8779 df-n0 8971 df-dec 9176 |
This theorem is referenced by: 9t9e81 9303 |
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