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Mirrors > Home > ILE Home > Th. List > 8p5e13 | GIF version |
Description: 8 + 5 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
8p5e13 | ⊢ (8 + 5) = ;13 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 8nn0 9216 | . 2 ⊢ 8 ∈ ℕ0 | |
2 | 4nn0 9212 | . 2 ⊢ 4 ∈ ℕ0 | |
3 | 2nn0 9210 | . 2 ⊢ 2 ∈ ℕ0 | |
4 | df-5 8998 | . 2 ⊢ 5 = (4 + 1) | |
5 | df-3 8996 | . 2 ⊢ 3 = (2 + 1) | |
6 | 8p4e12 9482 | . 2 ⊢ (8 + 4) = ;12 | |
7 | 1, 2, 3, 4, 5, 6 | 6p5lem 9470 | 1 ⊢ (8 + 5) = ;13 |
Colors of variables: wff set class |
Syntax hints: = wceq 1363 (class class class)co 5890 1c1 7829 + caddc 7831 2c2 8987 3c3 8988 4c4 8989 5c5 8990 8c8 8993 ;cdc 9401 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2162 ax-ext 2170 ax-sep 4135 ax-pow 4188 ax-pr 4223 ax-setind 4550 ax-cnex 7919 ax-resscn 7920 ax-1cn 7921 ax-1re 7922 ax-icn 7923 ax-addcl 7924 ax-addrcl 7925 ax-mulcl 7926 ax-addcom 7928 ax-mulcom 7929 ax-addass 7930 ax-mulass 7931 ax-distr 7932 ax-i2m1 7933 ax-1rid 7935 ax-0id 7936 ax-rnegex 7937 ax-cnre 7939 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2040 df-mo 2041 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ne 2360 df-ral 2472 df-rex 2473 df-reu 2474 df-rab 2476 df-v 2753 df-sbc 2977 df-dif 3145 df-un 3147 df-in 3149 df-ss 3156 df-pw 3591 df-sn 3612 df-pr 3613 df-op 3615 df-uni 3824 df-int 3859 df-br 4018 df-opab 4079 df-id 4307 df-xp 4646 df-rel 4647 df-cnv 4648 df-co 4649 df-dm 4650 df-iota 5192 df-fun 5232 df-fv 5238 df-riota 5846 df-ov 5893 df-oprab 5894 df-mpo 5895 df-sub 8147 df-inn 8937 df-2 8995 df-3 8996 df-4 8997 df-5 8998 df-6 8999 df-7 9000 df-8 9001 df-9 9002 df-n0 9194 df-dec 9402 |
This theorem is referenced by: 8p6e14 9484 |
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