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Mirrors > Home > MPE Home > Th. List > 7t3e21 | Structured version Visualization version GIF version |
Description: 7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
7t3e21 | ⊢ (7 · 3) = ;21 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 7nn0 11730 | . 2 ⊢ 7 ∈ ℕ0 | |
2 | 2nn0 11725 | . 2 ⊢ 2 ∈ ℕ0 | |
3 | df-3 11503 | . 2 ⊢ 3 = (2 + 1) | |
4 | 7t2e14 12021 | . 2 ⊢ (7 · 2) = ;14 | |
5 | 1nn0 11724 | . . 3 ⊢ 1 ∈ ℕ0 | |
6 | 4nn0 11727 | . . 3 ⊢ 4 ∈ ℕ0 | |
7 | eqid 2773 | . . 3 ⊢ ;14 = ;14 | |
8 | 1p1e2 11571 | . . 3 ⊢ (1 + 1) = 2 | |
9 | 1 | nn0cni 11719 | . . . 4 ⊢ 7 ∈ ℂ |
10 | 6 | nn0cni 11719 | . . . 4 ⊢ 4 ∈ ℂ |
11 | 7p4e11 11988 | . . . 4 ⊢ (7 + 4) = ;11 | |
12 | 9, 10, 11 | addcomli 10631 | . . 3 ⊢ (4 + 7) = ;11 |
13 | 5, 6, 1, 7, 8, 5, 12 | decaddci 11972 | . 2 ⊢ (;14 + 7) = ;21 |
14 | 1, 2, 3, 4, 13 | 4t3lem 12009 | 1 ⊢ (7 · 3) = ;21 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1508 (class class class)co 6975 1c1 10335 · cmul 10339 2c2 11494 3c3 11495 4c4 11496 7c7 11499 ;cdc 11910 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-er 8088 df-en 8306 df-dom 8307 df-sdom 8308 df-pnf 10475 df-mnf 10476 df-ltxr 10478 df-sub 10671 df-nn 11439 df-2 11502 df-3 11503 df-4 11504 df-5 11505 df-6 11506 df-7 11507 df-8 11508 df-9 11509 df-n0 11707 df-dec 11911 |
This theorem is referenced by: 7t4e28 12023 23prm 16307 prmlem2 16308 83prm 16311 163prm 16313 631prm 16315 1259prm 16324 log2ublem3 25244 log2ub 25245 ex-prmo 28032 hgt750lem2 31604 235t711 38643 ex-decpmul 38644 257prm 43121 |
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