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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno5lem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for fmtno5 43739. (Contributed by AV, 22-Jul-2021.) |
Ref | Expression |
---|---|
fmtno5lem3 | ⊢ (;;;;65536 · 3) = ;;;;;196608 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3nn0 11916 | . 2 ⊢ 3 ∈ ℕ0 | |
2 | 6nn0 11919 | . . . . 5 ⊢ 6 ∈ ℕ0 | |
3 | 5nn0 11918 | . . . . 5 ⊢ 5 ∈ ℕ0 | |
4 | 2, 3 | deccl 12114 | . . . 4 ⊢ ;65 ∈ ℕ0 |
5 | 4, 3 | deccl 12114 | . . 3 ⊢ ;;655 ∈ ℕ0 |
6 | 5, 1 | deccl 12114 | . 2 ⊢ ;;;6553 ∈ ℕ0 |
7 | eqid 2821 | . 2 ⊢ ;;;;65536 = ;;;;65536 | |
8 | 8nn0 11921 | . 2 ⊢ 8 ∈ ℕ0 | |
9 | 1nn0 11914 | . 2 ⊢ 1 ∈ ℕ0 | |
10 | 9nn0 11922 | . . . . . 6 ⊢ 9 ∈ ℕ0 | |
11 | 9, 10 | deccl 12114 | . . . . 5 ⊢ ;19 ∈ ℕ0 |
12 | 11, 2 | deccl 12114 | . . . 4 ⊢ ;;196 ∈ ℕ0 |
13 | 12, 3 | deccl 12114 | . . 3 ⊢ ;;;1965 ∈ ℕ0 |
14 | 5p1e6 11785 | . . . 4 ⊢ (5 + 1) = 6 | |
15 | eqid 2821 | . . . 4 ⊢ ;;;1965 = ;;;1965 | |
16 | 12, 3, 14, 15 | decsuc 12130 | . . 3 ⊢ (;;;1965 + 1) = ;;;1966 |
17 | eqid 2821 | . . . 4 ⊢ ;;;6553 = ;;;6553 | |
18 | eqid 2821 | . . . . 5 ⊢ ;;655 = ;;655 | |
19 | eqid 2821 | . . . . . . 7 ⊢ ;65 = ;65 | |
20 | 8p1e9 11788 | . . . . . . . 8 ⊢ (8 + 1) = 9 | |
21 | 6t3e18 12204 | . . . . . . . 8 ⊢ (6 · 3) = ;18 | |
22 | 9, 8, 20, 21 | decsuc 12130 | . . . . . . 7 ⊢ ((6 · 3) + 1) = ;19 |
23 | 5t3e15 12200 | . . . . . . 7 ⊢ (5 · 3) = ;15 | |
24 | 1, 2, 3, 19, 3, 9, 22, 23 | decmul1c 12164 | . . . . . 6 ⊢ (;65 · 3) = ;;195 |
25 | 11, 3, 14, 24 | decsuc 12130 | . . . . 5 ⊢ ((;65 · 3) + 1) = ;;196 |
26 | 1, 4, 3, 18, 3, 9, 25, 23 | decmul1c 12164 | . . . 4 ⊢ (;;655 · 3) = ;;;1965 |
27 | 3t3e9 11805 | . . . 4 ⊢ (3 · 3) = 9 | |
28 | 1, 5, 1, 17, 26, 27 | decmul1 12163 | . . 3 ⊢ (;;;6553 · 3) = ;;;;19659 |
29 | 13, 16, 28 | decsucc 12140 | . 2 ⊢ ((;;;6553 · 3) + 1) = ;;;;19660 |
30 | 1, 6, 2, 7, 8, 9, 29, 21 | decmul1c 12164 | 1 ⊢ (;;;;65536 · 3) = ;;;;;196608 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 (class class class)co 7156 0cc0 10537 1c1 10538 · cmul 10542 3c3 11694 5c5 11696 6c6 11697 8c8 11699 9c9 11700 ;cdc 12099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-sub 10872 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-dec 12100 |
This theorem is referenced by: fmtno5lem4 43738 |
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