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Mirrors > Home > ILE Home > Th. List > 8t5e40 | GIF version |
Description: 8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
8t5e40 | ⊢ (8 · 5) = ;40 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 8nn0 9197 | . 2 ⊢ 8 ∈ ℕ0 | |
2 | 4nn0 9193 | . 2 ⊢ 4 ∈ ℕ0 | |
3 | df-5 8979 | . 2 ⊢ 5 = (4 + 1) | |
4 | 8t4e32 9498 | . 2 ⊢ (8 · 4) = ;32 | |
5 | 3nn0 9192 | . . 3 ⊢ 3 ∈ ℕ0 | |
6 | 2nn0 9191 | . . 3 ⊢ 2 ∈ ℕ0 | |
7 | eqid 2177 | . . 3 ⊢ ;32 = ;32 | |
8 | 3p1e4 9052 | . . 3 ⊢ (3 + 1) = 4 | |
9 | 8cn 9003 | . . . 4 ⊢ 8 ∈ ℂ | |
10 | 2cn 8988 | . . . 4 ⊢ 2 ∈ ℂ | |
11 | 8p2e10 9461 | . . . 4 ⊢ (8 + 2) = ;10 | |
12 | 9, 10, 11 | addcomli 8100 | . . 3 ⊢ (2 + 8) = ;10 |
13 | 5, 6, 1, 7, 8, 12 | decaddci2 9443 | . 2 ⊢ (;32 + 8) = ;40 |
14 | 1, 2, 3, 4, 13 | 4t3lem 9478 | 1 ⊢ (8 · 5) = ;40 |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 (class class class)co 5874 0cc0 7810 1c1 7811 · cmul 7815 2c2 8968 3c3 8969 4c4 8970 5c5 8971 8c8 8974 ;cdc 9382 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-cnre 7921 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-sub 8128 df-inn 8918 df-2 8976 df-3 8977 df-4 8978 df-5 8979 df-6 8980 df-7 8981 df-8 8982 df-9 8983 df-n0 9175 df-dec 9383 |
This theorem is referenced by: 8t6e48 9500 |
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