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Mirrors > Home > ILE Home > Th. List > 8p7e15 | GIF version |
Description: 8 + 7 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
8p7e15 | ⊢ (8 + 7) = ;15 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 8nn0 8794 | . 2 ⊢ 8 ∈ ℕ0 | |
2 | 6nn0 8792 | . 2 ⊢ 6 ∈ ℕ0 | |
3 | 4nn0 8790 | . 2 ⊢ 4 ∈ ℕ0 | |
4 | df-7 8584 | . 2 ⊢ 7 = (6 + 1) | |
5 | df-5 8582 | . 2 ⊢ 5 = (4 + 1) | |
6 | 8p6e14 9059 | . 2 ⊢ (8 + 6) = ;14 | |
7 | 1, 2, 3, 4, 5, 6 | 6p5lem 9045 | 1 ⊢ (8 + 7) = ;15 |
Colors of variables: wff set class |
Syntax hints: = wceq 1296 (class class class)co 5690 1c1 7448 + caddc 7450 4c4 8573 5c5 8574 6c6 8575 7c7 8576 8c8 8577 ;cdc 8976 |
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-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-setind 4381 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-addcom 7542 ax-mulcom 7543 ax-addass 7544 ax-mulass 7545 ax-distr 7546 ax-i2m1 7547 ax-1rid 7549 ax-0id 7550 ax-rnegex 7551 ax-cnre 7553 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-br 3868 df-opab 3922 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-iota 5014 df-fun 5051 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-sub 7752 df-inn 8521 df-2 8579 df-3 8580 df-4 8581 df-5 8582 df-6 8583 df-7 8584 df-8 8585 df-9 8586 df-n0 8772 df-dec 8977 |
This theorem is referenced by: 8p8e16 9061 7t5e35 9087 |
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