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| Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno5lem1 | Structured version Visualization version GIF version | ||
| Description: Lemma 1 for fmtno5 47549. (Contributed by AV, 22-Jul-2021.) | 
| Ref | Expression | 
|---|---|
| fmtno5lem1 | ⊢ (;;;;65536 · 6) = ;;;;;393216 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 6nn0 12549 | . 2 ⊢ 6 ∈ ℕ0 | |
| 2 | 5nn0 12548 | . . . . 5 ⊢ 5 ∈ ℕ0 | |
| 3 | 1, 2 | deccl 12750 | . . . 4 ⊢ ;65 ∈ ℕ0 | 
| 4 | 3, 2 | deccl 12750 | . . 3 ⊢ ;;655 ∈ ℕ0 | 
| 5 | 3nn0 12546 | . . 3 ⊢ 3 ∈ ℕ0 | |
| 6 | 4, 5 | deccl 12750 | . 2 ⊢ ;;;6553 ∈ ℕ0 | 
| 7 | eqid 2736 | . 2 ⊢ ;;;;65536 = ;;;;65536 | |
| 8 | 9nn0 12552 | . . . . . 6 ⊢ 9 ∈ ℕ0 | |
| 9 | 5, 8 | deccl 12750 | . . . . 5 ⊢ ;39 ∈ ℕ0 | 
| 10 | 9, 5 | deccl 12750 | . . . 4 ⊢ ;;393 ∈ ℕ0 | 
| 11 | 1nn0 12544 | . . . 4 ⊢ 1 ∈ ℕ0 | |
| 12 | 10, 11 | deccl 12750 | . . 3 ⊢ ;;;3931 ∈ ℕ0 | 
| 13 | 8nn0 12551 | . . 3 ⊢ 8 ∈ ℕ0 | |
| 14 | eqid 2736 | . . . 4 ⊢ ;;;6553 = ;;;6553 | |
| 15 | 0nn0 12543 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
| 16 | 0p1e1 12389 | . . . . 5 ⊢ (0 + 1) = 1 | |
| 17 | eqid 2736 | . . . . . 6 ⊢ ;;655 = ;;655 | |
| 18 | eqid 2736 | . . . . . . . 8 ⊢ ;65 = ;65 | |
| 19 | 6t6e36 12843 | . . . . . . . . 9 ⊢ (6 · 6) = ;36 | |
| 20 | 6p3e9 12427 | . . . . . . . . 9 ⊢ (6 + 3) = 9 | |
| 21 | 5, 1, 5, 19, 20 | decaddi 12795 | . . . . . . . 8 ⊢ ((6 · 6) + 3) = ;39 | 
| 22 | 6cn 12358 | . . . . . . . . 9 ⊢ 6 ∈ ℂ | |
| 23 | 5cn 12355 | . . . . . . . . 9 ⊢ 5 ∈ ℂ | |
| 24 | 6t5e30 12842 | . . . . . . . . 9 ⊢ (6 · 5) = ;30 | |
| 25 | 22, 23, 24 | mulcomli 11271 | . . . . . . . 8 ⊢ (5 · 6) = ;30 | 
| 26 | 1, 1, 2, 18, 15, 5, 21, 25 | decmul1c 12800 | . . . . . . 7 ⊢ (;65 · 6) = ;;390 | 
| 27 | 3cn 12348 | . . . . . . . 8 ⊢ 3 ∈ ℂ | |
| 28 | 27 | addlidi 11450 | . . . . . . 7 ⊢ (0 + 3) = 3 | 
| 29 | 9, 15, 5, 26, 28 | decaddi 12795 | . . . . . 6 ⊢ ((;65 · 6) + 3) = ;;393 | 
| 30 | 1, 3, 2, 17, 15, 5, 29, 25 | decmul1c 12800 | . . . . 5 ⊢ (;;655 · 6) = ;;;3930 | 
| 31 | 10, 15, 16, 30 | decsuc 12766 | . . . 4 ⊢ ((;;655 · 6) + 1) = ;;;3931 | 
| 32 | 6t3e18 12840 | . . . . 5 ⊢ (6 · 3) = ;18 | |
| 33 | 22, 27, 32 | mulcomli 11271 | . . . 4 ⊢ (3 · 6) = ;18 | 
| 34 | 1, 4, 5, 14, 13, 11, 31, 33 | decmul1c 12800 | . . 3 ⊢ (;;;6553 · 6) = ;;;;39318 | 
| 35 | 1p1e2 12392 | . . . 4 ⊢ (1 + 1) = 2 | |
| 36 | eqid 2736 | . . . 4 ⊢ ;;;3931 = ;;;3931 | |
| 37 | 10, 11, 35, 36 | decsuc 12766 | . . 3 ⊢ (;;;3931 + 1) = ;;;3932 | 
| 38 | 8p3e11 12816 | . . 3 ⊢ (8 + 3) = ;11 | |
| 39 | 12, 13, 5, 34, 37, 11, 38 | decaddci 12796 | . 2 ⊢ ((;;;6553 · 6) + 3) = ;;;;39321 | 
| 40 | 1, 6, 1, 7, 1, 5, 39, 19 | decmul1c 12800 | 1 ⊢ (;;;;65536 · 6) = ;;;;;393216 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 (class class class)co 7432 0cc0 11156 1c1 11157 · cmul 11161 2c2 12322 3c3 12323 5c5 12325 6c6 12326 8c8 12328 9c9 12329 ;cdc 12735 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-ltxr 11301 df-sub 11495 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-dec 12736 | 
| This theorem is referenced by: fmtno5lem4 47548 | 
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