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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno5lem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for fmtno5 42490. (Contributed by AV, 22-Jul-2021.) |
Ref | Expression |
---|---|
fmtno5lem1 | ⊢ (;;;;65536 · 6) = ;;;;;393216 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6nn0 11665 | . 2 ⊢ 6 ∈ ℕ0 | |
2 | 5nn0 11664 | . . . . 5 ⊢ 5 ∈ ℕ0 | |
3 | 1, 2 | deccl 11860 | . . . 4 ⊢ ;65 ∈ ℕ0 |
4 | 3, 2 | deccl 11860 | . . 3 ⊢ ;;655 ∈ ℕ0 |
5 | 3nn0 11662 | . . 3 ⊢ 3 ∈ ℕ0 | |
6 | 4, 5 | deccl 11860 | . 2 ⊢ ;;;6553 ∈ ℕ0 |
7 | eqid 2778 | . 2 ⊢ ;;;;65536 = ;;;;65536 | |
8 | 9nn0 11668 | . . . . . 6 ⊢ 9 ∈ ℕ0 | |
9 | 5, 8 | deccl 11860 | . . . . 5 ⊢ ;39 ∈ ℕ0 |
10 | 9, 5 | deccl 11860 | . . . 4 ⊢ ;;393 ∈ ℕ0 |
11 | 1nn0 11660 | . . . 4 ⊢ 1 ∈ ℕ0 | |
12 | 10, 11 | deccl 11860 | . . 3 ⊢ ;;;3931 ∈ ℕ0 |
13 | 8nn0 11667 | . . 3 ⊢ 8 ∈ ℕ0 | |
14 | eqid 2778 | . . . 4 ⊢ ;;;6553 = ;;;6553 | |
15 | 0nn0 11659 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
16 | 0p1e1 11504 | . . . . 5 ⊢ (0 + 1) = 1 | |
17 | eqid 2778 | . . . . . 6 ⊢ ;;655 = ;;655 | |
18 | eqid 2778 | . . . . . . . 8 ⊢ ;65 = ;65 | |
19 | 6t6e36 11955 | . . . . . . . . 9 ⊢ (6 · 6) = ;36 | |
20 | 6p3e9 11542 | . . . . . . . . 9 ⊢ (6 + 3) = 9 | |
21 | 5, 1, 5, 19, 20 | decaddi 11906 | . . . . . . . 8 ⊢ ((6 · 6) + 3) = ;39 |
22 | 6cn 11469 | . . . . . . . . 9 ⊢ 6 ∈ ℂ | |
23 | 5cn 11465 | . . . . . . . . 9 ⊢ 5 ∈ ℂ | |
24 | 6t5e30 11954 | . . . . . . . . 9 ⊢ (6 · 5) = ;30 | |
25 | 22, 23, 24 | mulcomli 10386 | . . . . . . . 8 ⊢ (5 · 6) = ;30 |
26 | 1, 1, 2, 18, 15, 5, 21, 25 | decmul1c 11912 | . . . . . . 7 ⊢ (;65 · 6) = ;;390 |
27 | 3cn 11456 | . . . . . . . 8 ⊢ 3 ∈ ℂ | |
28 | 27 | addid2i 10564 | . . . . . . 7 ⊢ (0 + 3) = 3 |
29 | 9, 15, 5, 26, 28 | decaddi 11906 | . . . . . 6 ⊢ ((;65 · 6) + 3) = ;;393 |
30 | 1, 3, 2, 17, 15, 5, 29, 25 | decmul1c 11912 | . . . . 5 ⊢ (;;655 · 6) = ;;;3930 |
31 | 10, 15, 16, 30 | decsuc 11877 | . . . 4 ⊢ ((;;655 · 6) + 1) = ;;;3931 |
32 | 6t3e18 11952 | . . . . 5 ⊢ (6 · 3) = ;18 | |
33 | 22, 27, 32 | mulcomli 10386 | . . . 4 ⊢ (3 · 6) = ;18 |
34 | 1, 4, 5, 14, 13, 11, 31, 33 | decmul1c 11912 | . . 3 ⊢ (;;;6553 · 6) = ;;;;39318 |
35 | 1p1e2 11507 | . . . 4 ⊢ (1 + 1) = 2 | |
36 | eqid 2778 | . . . 4 ⊢ ;;;3931 = ;;;3931 | |
37 | 10, 11, 35, 36 | decsuc 11877 | . . 3 ⊢ (;;;3931 + 1) = ;;;3932 |
38 | 8p3e11 11928 | . . 3 ⊢ (8 + 3) = ;11 | |
39 | 12, 13, 5, 34, 37, 11, 38 | decaddci 11907 | . 2 ⊢ ((;;;6553 · 6) + 3) = ;;;;39321 |
40 | 1, 6, 1, 7, 1, 5, 39, 19 | decmul1c 11912 | 1 ⊢ (;;;;65536 · 6) = ;;;;;393216 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 (class class class)co 6922 0cc0 10272 1c1 10273 · cmul 10277 2c2 11430 3c3 11431 5c5 11433 6c6 11434 8c8 11436 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: fmtno5lem4 42489 |
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