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| Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno5lem3 | Structured version Visualization version GIF version | ||
| Description: Lemma 3 for fmtno5 47544. (Contributed by AV, 22-Jul-2021.) |
| Ref | Expression |
|---|---|
| fmtno5lem3 | ⊢ (;;;;65536 · 3) = ;;;;;196608 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 3nn0 12544 | . 2 ⊢ 3 ∈ ℕ0 | |
| 2 | 6nn0 12547 | . . . . 5 ⊢ 6 ∈ ℕ0 | |
| 3 | 5nn0 12546 | . . . . 5 ⊢ 5 ∈ ℕ0 | |
| 4 | 2, 3 | deccl 12748 | . . . 4 ⊢ ;65 ∈ ℕ0 |
| 5 | 4, 3 | deccl 12748 | . . 3 ⊢ ;;655 ∈ ℕ0 |
| 6 | 5, 1 | deccl 12748 | . 2 ⊢ ;;;6553 ∈ ℕ0 |
| 7 | eqid 2737 | . 2 ⊢ ;;;;65536 = ;;;;65536 | |
| 8 | 8nn0 12549 | . 2 ⊢ 8 ∈ ℕ0 | |
| 9 | 1nn0 12542 | . 2 ⊢ 1 ∈ ℕ0 | |
| 10 | 9nn0 12550 | . . . . . 6 ⊢ 9 ∈ ℕ0 | |
| 11 | 9, 10 | deccl 12748 | . . . . 5 ⊢ ;19 ∈ ℕ0 |
| 12 | 11, 2 | deccl 12748 | . . . 4 ⊢ ;;196 ∈ ℕ0 |
| 13 | 12, 3 | deccl 12748 | . . 3 ⊢ ;;;1965 ∈ ℕ0 |
| 14 | 5p1e6 12413 | . . . 4 ⊢ (5 + 1) = 6 | |
| 15 | eqid 2737 | . . . 4 ⊢ ;;;1965 = ;;;1965 | |
| 16 | 12, 3, 14, 15 | decsuc 12764 | . . 3 ⊢ (;;;1965 + 1) = ;;;1966 |
| 17 | eqid 2737 | . . . 4 ⊢ ;;;6553 = ;;;6553 | |
| 18 | eqid 2737 | . . . . 5 ⊢ ;;655 = ;;655 | |
| 19 | eqid 2737 | . . . . . . 7 ⊢ ;65 = ;65 | |
| 20 | 8p1e9 12416 | . . . . . . . 8 ⊢ (8 + 1) = 9 | |
| 21 | 6t3e18 12838 | . . . . . . . 8 ⊢ (6 · 3) = ;18 | |
| 22 | 9, 8, 20, 21 | decsuc 12764 | . . . . . . 7 ⊢ ((6 · 3) + 1) = ;19 |
| 23 | 5t3e15 12834 | . . . . . . 7 ⊢ (5 · 3) = ;15 | |
| 24 | 1, 2, 3, 19, 3, 9, 22, 23 | decmul1c 12798 | . . . . . 6 ⊢ (;65 · 3) = ;;195 |
| 25 | 11, 3, 14, 24 | decsuc 12764 | . . . . 5 ⊢ ((;65 · 3) + 1) = ;;196 |
| 26 | 1, 4, 3, 18, 3, 9, 25, 23 | decmul1c 12798 | . . . 4 ⊢ (;;655 · 3) = ;;;1965 |
| 27 | 3t3e9 12433 | . . . 4 ⊢ (3 · 3) = 9 | |
| 28 | 1, 5, 1, 17, 26, 27 | decmul1 12797 | . . 3 ⊢ (;;;6553 · 3) = ;;;;19659 |
| 29 | 13, 16, 28 | decsucc 12774 | . 2 ⊢ ((;;;6553 · 3) + 1) = ;;;;19660 |
| 30 | 1, 6, 2, 7, 8, 9, 29, 21 | decmul1c 12798 | 1 ⊢ (;;;;65536 · 3) = ;;;;;196608 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 (class class class)co 7431 0cc0 11155 1c1 11156 · cmul 11160 3c3 12322 5c5 12324 6c6 12325 8c8 12327 9c9 12328 ;cdc 12733 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-sub 11494 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-dec 12734 |
| This theorem is referenced by: fmtno5lem4 47543 |
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