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Mirrors > Home > ILE Home > Th. List > 7t7e49 | GIF version |
Description: 7 times 7 equals 49. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
7t7e49 | ⊢ (7 · 7) = ;49 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 7nn0 9144 | . 2 ⊢ 7 ∈ ℕ0 | |
2 | 6nn0 9143 | . 2 ⊢ 6 ∈ ℕ0 | |
3 | df-7 8929 | . 2 ⊢ 7 = (6 + 1) | |
4 | 7t6e42 9442 | . 2 ⊢ (7 · 6) = ;42 | |
5 | 4nn0 9141 | . . 3 ⊢ 4 ∈ ℕ0 | |
6 | 2nn0 9139 | . . 3 ⊢ 2 ∈ ℕ0 | |
7 | eqid 2170 | . . 3 ⊢ ;42 = ;42 | |
8 | 7cn 8949 | . . . 4 ⊢ 7 ∈ ℂ | |
9 | 2cn 8936 | . . . 4 ⊢ 2 ∈ ℂ | |
10 | 7p2e9 9016 | . . . 4 ⊢ (7 + 2) = 9 | |
11 | 8, 9, 10 | addcomli 8051 | . . 3 ⊢ (2 + 7) = 9 |
12 | 5, 6, 1, 7, 11 | decaddi 9389 | . 2 ⊢ (;42 + 7) = ;49 |
13 | 1, 2, 3, 4, 12 | 4t3lem 9426 | 1 ⊢ (7 · 7) = ;49 |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 (class class class)co 5850 · cmul 7766 2c2 8916 4c4 8918 6c6 8920 7c7 8921 9c9 8923 ;cdc 9330 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-setind 4519 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-addcom 7861 ax-mulcom 7862 ax-addass 7863 ax-mulass 7864 ax-distr 7865 ax-i2m1 7866 ax-1rid 7868 ax-0id 7869 ax-rnegex 7870 ax-cnre 7872 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-sub 8079 df-inn 8866 df-2 8924 df-3 8925 df-4 8926 df-5 8927 df-6 8928 df-7 8929 df-8 8930 df-9 8931 df-n0 9123 df-dec 9331 |
This theorem is referenced by: (None) |
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