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Mirrors > Home > MPE Home > Th. List > 9t6e54 | Structured version Visualization version 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 11668 | . 2 ⊢ 9 ∈ ℕ0 | |
2 | 5nn0 11664 | . 2 ⊢ 5 ∈ ℕ0 | |
3 | df-6 11442 | . 2 ⊢ 6 = (5 + 1) | |
4 | 9t5e45 11972 | . 2 ⊢ (9 · 5) = ;45 | |
5 | 4nn0 11663 | . . 3 ⊢ 4 ∈ ℕ0 | |
6 | eqid 2778 | . . 3 ⊢ ;45 = ;45 | |
7 | 4p1e5 11528 | . . 3 ⊢ (4 + 1) = 5 | |
8 | 1 | nn0cni 11655 | . . . 4 ⊢ 9 ∈ ℂ |
9 | 2 | nn0cni 11655 | . . . 4 ⊢ 5 ∈ ℂ |
10 | 9p5e14 11937 | . . . 4 ⊢ (9 + 5) = ;14 | |
11 | 8, 9, 10 | addcomli 10568 | . . 3 ⊢ (5 + 9) = ;14 |
12 | 5, 2, 1, 6, 7, 5, 11 | decaddci 11907 | . 2 ⊢ (;45 + 9) = ;54 |
13 | 1, 2, 3, 4, 12 | 4t3lem 11944 | 1 ⊢ (9 · 6) = ;54 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 (class class class)co 6922 1c1 10273 · cmul 10277 4c4 11432 5c5 11433 6c6 11434 9c9 11437 ;cdc 11845 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-ltxr 10416 df-sub 10608 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-dec 11846 |
This theorem is referenced by: 9t7e63 11974 317prm 16231 hgt750lem2 31332 127prm 42536 |
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