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Mirrors > Home > ILE Home > Th. List > 7t3e21 | Unicode version |
Description: 7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
7t3e21 | ; |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 7nn0 8967 | . 2 | |
2 | 2nn0 8962 | . 2 | |
3 | df-3 8748 | . 2 | |
4 | 7t2e14 9258 | . 2 ; | |
5 | 1nn0 8961 | . . 3 | |
6 | 4nn0 8964 | . . 3 | |
7 | eqid 2117 | . . 3 ; ; | |
8 | 1p1e2 8805 | . . 3 | |
9 | 1 | nn0cni 8957 | . . . 4 |
10 | 6 | nn0cni 8957 | . . . 4 |
11 | 7p4e11 9225 | . . . 4 ; | |
12 | 9, 10, 11 | addcomli 7875 | . . 3 ; |
13 | 5, 6, 1, 7, 8, 5, 12 | decaddci 9210 | . 2 ; ; |
14 | 1, 2, 3, 4, 13 | 4t3lem 9246 | 1 ; |
Colors of variables: wff set class |
Syntax hints: wceq 1316 (class class class)co 5742 c1 7589 cmul 7593 c2 8739 c3 8740 c4 8741 c7 8744 ;cdc 9150 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-sub 7903 df-inn 8689 df-2 8747 df-3 8748 df-4 8749 df-5 8750 df-6 8751 df-7 8752 df-8 8753 df-9 8754 df-n0 8946 df-dec 9151 |
This theorem is referenced by: 7t4e28 9260 |
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