![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > 5t3e15 | GIF version |
Description: 5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
5t3e15 | ⊢ (5 · 3) = ;15 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 5nn0 8747 | . 2 ⊢ 5 ∈ ℕ0 | |
2 | 2nn0 8744 | . 2 ⊢ 2 ∈ ℕ0 | |
3 | df-3 8536 | . 2 ⊢ 3 = (2 + 1) | |
4 | 5t2e10 9030 | . 2 ⊢ (5 · 2) = ;10 | |
5 | dec10p 8973 | . 2 ⊢ (;10 + 5) = ;15 | |
6 | 1, 2, 3, 4, 5 | 4t3lem 9027 | 1 ⊢ (5 · 3) = ;15 |
Colors of variables: wff set class |
Syntax hints: = wceq 1290 (class class class)co 5666 0cc0 7404 1c1 7405 · cmul 7409 2c2 8527 3c3 8528 5c5 8530 ;cdc 8931 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-cnex 7490 ax-resscn 7491 ax-1cn 7492 ax-1re 7493 ax-icn 7494 ax-addcl 7495 ax-addrcl 7496 ax-mulcl 7497 ax-mulcom 7500 ax-addass 7501 ax-mulass 7502 ax-distr 7503 ax-1rid 7506 ax-0id 7507 ax-rnegex 7508 ax-cnre 7510 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-rab 2369 df-v 2622 df-un 3004 df-in 3006 df-ss 3013 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-br 3852 df-iota 4993 df-fv 5036 df-ov 5669 df-inn 8477 df-2 8535 df-3 8536 df-4 8537 df-5 8538 df-6 8539 df-7 8540 df-8 8541 df-9 8542 df-n0 8728 df-dec 8932 |
This theorem is referenced by: 5t4e20 9032 |
Copyright terms: Public domain | W3C validator |