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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 9231 | . 2 ⊢ 9 ∈ ℕ0 | |
2 | 6nn0 9228 | . 2 ⊢ 6 ∈ ℕ0 | |
3 | df-7 9014 | . 2 ⊢ 7 = (6 + 1) | |
4 | 9t6e54 9540 | . 2 ⊢ (9 · 6) = ;54 | |
5 | 5nn0 9227 | . . 3 ⊢ 5 ∈ ℕ0 | |
6 | 4nn0 9226 | . . 3 ⊢ 4 ∈ ℕ0 | |
7 | eqid 2189 | . . 3 ⊢ ;54 = ;54 | |
8 | 5p1e6 9087 | . . 3 ⊢ (5 + 1) = 6 | |
9 | 3nn0 9225 | . . 3 ⊢ 3 ∈ ℕ0 | |
10 | 1 | nn0cni 9219 | . . . 4 ⊢ 9 ∈ ℂ |
11 | 6 | nn0cni 9219 | . . . 4 ⊢ 4 ∈ ℂ |
12 | 9p4e13 9503 | . . . 4 ⊢ (9 + 4) = ;13 | |
13 | 10, 11, 12 | addcomli 8133 | . . 3 ⊢ (4 + 9) = ;13 |
14 | 5, 6, 1, 7, 8, 9, 13 | decaddci 9475 | . 2 ⊢ (;54 + 9) = ;63 |
15 | 1, 2, 3, 4, 14 | 4t3lem 9511 | 1 ⊢ (9 · 7) = ;63 |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 (class class class)co 5897 1c1 7843 · cmul 7847 3c3 9002 4c4 9003 5c5 9004 6c6 9005 7c7 9006 9c9 9008 ;cdc 9415 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-addcom 7942 ax-mulcom 7943 ax-addass 7944 ax-mulass 7945 ax-distr 7946 ax-i2m1 7947 ax-1rid 7949 ax-0id 7950 ax-rnegex 7951 ax-cnre 7953 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-sub 8161 df-inn 8951 df-2 9009 df-3 9010 df-4 9011 df-5 9012 df-6 9013 df-7 9014 df-8 9015 df-9 9016 df-n0 9208 df-dec 9416 |
This theorem is referenced by: 9t8e72 9542 |
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