Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 8205 | . 2 ⊢ 9 ∈ ℕ0 | |
2 | 6nn0 8202 | . 2 ⊢ 6 ∈ ℕ0 | |
3 | df-7 7978 | . 2 ⊢ 7 = (6 + 1) | |
4 | 9t6e54 8466 | . 2 ⊢ (9 · 6) = ;54 | |
5 | 5nn0 8201 | . . 3 ⊢ 5 ∈ ℕ0 | |
6 | 4nn0 8200 | . . 3 ⊢ 4 ∈ ℕ0 | |
7 | eqid 2040 | . . 3 ⊢ ;54 = ;54 | |
8 | 5p1e6 8047 | . . 3 ⊢ (5 + 1) = 6 | |
9 | 3nn0 8199 | . . 3 ⊢ 3 ∈ ℕ0 | |
10 | 1 | nn0cni 8193 | . . . 4 ⊢ 9 ∈ ℂ |
11 | 6 | nn0cni 8193 | . . . 4 ⊢ 4 ∈ ℂ |
12 | 9p4e13 8431 | . . . 4 ⊢ (9 + 4) = ;13 | |
13 | 10, 11, 12 | addcomli 7158 | . . 3 ⊢ (4 + 9) = ;13 |
14 | 5, 6, 1, 7, 8, 9, 13 | decaddci 8412 | . 2 ⊢ (;54 + 9) = ;63 |
15 | 1, 2, 3, 4, 14 | 4t3lem 8438 | 1 ⊢ (9 · 7) = ;63 |
Colors of variables: wff set class |
Syntax hints: = wceq 1243 (class class class)co 5512 1c1 6890 · cmul 6894 3c3 7965 4c4 7966 5c5 7967 6c6 7968 7c7 7969 9c9 7971 ;cdc 8368 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-mulcom 6985 ax-addass 6986 ax-mulass 6987 ax-distr 6988 ax-i2m1 6989 ax-1rid 6991 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-sub 7184 df-inn 7915 df-2 7973 df-3 7974 df-4 7975 df-5 7976 df-6 7977 df-7 7978 df-8 7979 df-9 7980 df-10 7981 df-n0 8182 df-dec 8369 |
This theorem is referenced by: 9t8e72 8468 |
Copyright terms: Public domain | W3C validator |