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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 12421 | . . . 4 ⊢ (2 · 2) = 4 | |
2 | 1 | eqcomi 2735 | . . 3 ⊢ 4 = (2 · 2) |
3 | 2 | oveq1i 7425 | . 2 ⊢ (4↑2) = ((2 · 2)↑2) |
4 | 2cn 12332 | . . 3 ⊢ 2 ∈ ℂ | |
5 | 4, 4 | sqmuli 14195 | . 2 ⊢ ((2 · 2)↑2) = ((2↑2) · (2↑2)) |
6 | 4 | sqvali 14191 | . . . 4 ⊢ (2↑2) = (2 · 2) |
7 | sq2 14208 | . . . 4 ⊢ (2↑2) = 4 | |
8 | 6, 7 | oveq12i 7427 | . . 3 ⊢ ((2↑2) · (2↑2)) = ((2 · 2) · 4) |
9 | 4cn 12342 | . . . 4 ⊢ 4 ∈ ℂ | |
10 | 4, 4, 9 | mulassi 11265 | . . 3 ⊢ ((2 · 2) · 4) = (2 · (2 · 4)) |
11 | 4t2e8 12425 | . . . . 5 ⊢ (4 · 2) = 8 | |
12 | 9, 4, 11 | mulcomli 11263 | . . . 4 ⊢ (2 · 4) = 8 |
13 | 12 | oveq2i 7426 | . . 3 ⊢ (2 · (2 · 4)) = (2 · 8) |
14 | 8, 10, 13 | 3eqtri 2758 | . 2 ⊢ ((2↑2) · (2↑2)) = (2 · 8) |
15 | 3, 5, 14 | 3eqtri 2758 | 1 ⊢ (4↑2) = (2 · 8) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 (class class class)co 7415 · cmul 11153 2c2 12312 4c4 12314 8c8 12318 ↑cexp 14074 |
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 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7737 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3366 df-rab 3421 df-v 3466 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3968 df-nul 4325 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4908 df-iun 4997 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6497 df-fun 6547 df-fn 6548 df-f 6549 df-f1 6550 df-fo 6551 df-f1o 6552 df-fv 6553 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8966 df-dom 8967 df-sdom 8968 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12258 df-2 12320 df-3 12321 df-4 12322 df-5 12323 df-6 12324 df-7 12325 df-8 12326 df-n0 12518 df-z 12604 df-uz 12868 df-seq 14015 df-exp 14075 |
This theorem is referenced by: 2lgsoddprmlem3c 27437 2lgsoddprmlem3d 27438 ex-exp 30379 |
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