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Mirrors > Home > MPE Home > Th. List > 9t9e81 | Structured version Visualization version GIF version |
Description: 9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
9t9e81 | ⊢ (9 · 9) = ;81 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 9nn0 12526 | . 2 ⊢ 9 ∈ ℕ0 | |
2 | 8nn0 12525 | . 2 ⊢ 8 ∈ ℕ0 | |
3 | df-9 12312 | . 2 ⊢ 9 = (8 + 1) | |
4 | 9t8e72 12835 | . 2 ⊢ (9 · 8) = ;72 | |
5 | 7nn0 12524 | . . 3 ⊢ 7 ∈ ℕ0 | |
6 | 2nn0 12519 | . . 3 ⊢ 2 ∈ ℕ0 | |
7 | eqid 2728 | . . 3 ⊢ ;72 = ;72 | |
8 | 7p1e8 12391 | . . 3 ⊢ (7 + 1) = 8 | |
9 | 1nn0 12518 | . . 3 ⊢ 1 ∈ ℕ0 | |
10 | 9cn 12342 | . . . 4 ⊢ 9 ∈ ℂ | |
11 | 2cn 12317 | . . . 4 ⊢ 2 ∈ ℂ | |
12 | 9p2e11 12794 | . . . 4 ⊢ (9 + 2) = ;11 | |
13 | 10, 11, 12 | addcomli 11436 | . . 3 ⊢ (2 + 9) = ;11 |
14 | 5, 6, 1, 7, 8, 9, 13 | decaddci 12768 | . 2 ⊢ (;72 + 9) = ;81 |
15 | 1, 2, 3, 4, 14 | 4t3lem 12804 | 1 ⊢ (9 · 9) = ;81 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 (class class class)co 7420 1c1 11139 · cmul 11143 2c2 12297 7c7 12302 8c8 12303 9c9 12304 ;cdc 12707 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-ltxr 11283 df-sub 11476 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-dec 12708 |
This theorem is referenced by: prmlem2 17088 2503lem2 17106 4001lem1 17109 4001lem2 17110 log2ublem3 26879 hgt750lem2 34284 3lexlogpow5ineq1 41525 3lexlogpow5ineq5 41531 sq9 41866 fmtno4nprmfac193 46914 3exp4mod41 46956 |
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