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Mirrors > Home > MPE Home > Th. List > sq4e2t8 | Structured version Visualization version GIF version |
Description: The square of 4 is 2 times 8. (Contributed by AV, 20-Jul-2021.) |
Ref | Expression |
---|---|
sq4e2t8 | ⊢ (4↑2) = (2 · 8) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2t2e4 11483 | . . . 4 ⊢ (2 · 2) = 4 | |
2 | 1 | eqcomi 2809 | . . 3 ⊢ 4 = (2 · 2) |
3 | 2 | oveq1i 6889 | . 2 ⊢ (4↑2) = ((2 · 2)↑2) |
4 | 2cn 11387 | . . 3 ⊢ 2 ∈ ℂ | |
5 | 4, 4 | sqmuli 13200 | . 2 ⊢ ((2 · 2)↑2) = ((2↑2) · (2↑2)) |
6 | 4 | sqvali 13196 | . . . 4 ⊢ (2↑2) = (2 · 2) |
7 | sq2 13213 | . . . 4 ⊢ (2↑2) = 4 | |
8 | 6, 7 | oveq12i 6891 | . . 3 ⊢ ((2↑2) · (2↑2)) = ((2 · 2) · 4) |
9 | 4cn 11398 | . . . 4 ⊢ 4 ∈ ℂ | |
10 | 4, 4, 9 | mulassi 10341 | . . 3 ⊢ ((2 · 2) · 4) = (2 · (2 · 4)) |
11 | 4t2e8 11487 | . . . . 5 ⊢ (4 · 2) = 8 | |
12 | 9, 4, 11 | mulcomli 10339 | . . . 4 ⊢ (2 · 4) = 8 |
13 | 12 | oveq2i 6890 | . . 3 ⊢ (2 · (2 · 4)) = (2 · 8) |
14 | 8, 10, 13 | 3eqtri 2826 | . 2 ⊢ ((2↑2) · (2↑2)) = (2 · 8) |
15 | 3, 5, 14 | 3eqtri 2826 | 1 ⊢ (4↑2) = (2 · 8) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1653 (class class class)co 6879 · cmul 10230 2c2 11367 4c4 11369 8c8 11373 ↑cexp 13113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 ax-cnex 10281 ax-resscn 10282 ax-1cn 10283 ax-icn 10284 ax-addcl 10285 ax-addrcl 10286 ax-mulcl 10287 ax-mulrcl 10288 ax-mulcom 10289 ax-addass 10290 ax-mulass 10291 ax-distr 10292 ax-i2m1 10293 ax-1ne0 10294 ax-1rid 10295 ax-rnegex 10296 ax-rrecex 10297 ax-cnre 10298 ax-pre-lttri 10299 ax-pre-lttrn 10300 ax-pre-ltadd 10301 ax-pre-mulgt0 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-pss 3786 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-tp 4374 df-op 4376 df-uni 4630 df-iun 4713 df-br 4845 df-opab 4907 df-mpt 4924 df-tr 4947 df-id 5221 df-eprel 5226 df-po 5234 df-so 5235 df-fr 5272 df-we 5274 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-pred 5899 df-ord 5945 df-on 5946 df-lim 5947 df-suc 5948 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-fv 6110 df-riota 6840 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-om 7301 df-2nd 7403 df-wrecs 7646 df-recs 7708 df-rdg 7746 df-er 7983 df-en 8197 df-dom 8198 df-sdom 8199 df-pnf 10366 df-mnf 10367 df-xr 10368 df-ltxr 10369 df-le 10370 df-sub 10559 df-neg 10560 df-nn 11314 df-2 11375 df-3 11376 df-4 11377 df-5 11378 df-6 11379 df-7 11380 df-8 11381 df-n0 11580 df-z 11666 df-uz 11930 df-seq 13055 df-exp 13114 |
This theorem is referenced by: 2lgsoddprmlem3c 25488 2lgsoddprmlem3d 25489 ex-exp 27834 |
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