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Mirrors > Home > ILE Home > Th. List > 9t6e54 | GIF version |
Description: 9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
9t6e54 | ⊢ (9 · 6) = ;54 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 9nn0 9267 | . 2 ⊢ 9 ∈ ℕ0 | |
2 | 5nn0 9263 | . 2 ⊢ 5 ∈ ℕ0 | |
3 | df-6 9047 | . 2 ⊢ 6 = (5 + 1) | |
4 | 9t5e45 9575 | . 2 ⊢ (9 · 5) = ;45 | |
5 | 4nn0 9262 | . . 3 ⊢ 4 ∈ ℕ0 | |
6 | eqid 2193 | . . 3 ⊢ ;45 = ;45 | |
7 | 4p1e5 9121 | . . 3 ⊢ (4 + 1) = 5 | |
8 | 1 | nn0cni 9255 | . . . 4 ⊢ 9 ∈ ℂ |
9 | 2 | nn0cni 9255 | . . . 4 ⊢ 5 ∈ ℂ |
10 | 9p5e14 9540 | . . . 4 ⊢ (9 + 5) = ;14 | |
11 | 8, 9, 10 | addcomli 8166 | . . 3 ⊢ (5 + 9) = ;14 |
12 | 5, 2, 1, 6, 7, 5, 11 | decaddci 9511 | . 2 ⊢ (;45 + 9) = ;54 |
13 | 1, 2, 3, 4, 12 | 4t3lem 9547 | 1 ⊢ (9 · 6) = ;54 |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 (class class class)co 5919 1c1 7875 · cmul 7879 4c4 9037 5c5 9038 6c6 9039 9c9 9042 ;cdc 9451 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-setind 4570 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-cnre 7985 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2987 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-br 4031 df-opab 4092 df-id 4325 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-iota 5216 df-fun 5257 df-fv 5263 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-sub 8194 df-inn 8985 df-2 9043 df-3 9044 df-4 9045 df-5 9046 df-6 9047 df-7 9048 df-8 9049 df-9 9050 df-n0 9244 df-dec 9452 |
This theorem is referenced by: 9t7e63 9577 |
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