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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno5lem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for fmtno5 43713. (Contributed by AV, 22-Jul-2021.) |
Ref | Expression |
---|---|
fmtno5lem2 | ⊢ (;;;;65536 · 5) = ;;;;;327680 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 5nn0 11911 | . 2 ⊢ 5 ∈ ℕ0 | |
2 | 6nn0 11912 | . . . . 5 ⊢ 6 ∈ ℕ0 | |
3 | 2, 1 | deccl 12107 | . . . 4 ⊢ ;65 ∈ ℕ0 |
4 | 3, 1 | deccl 12107 | . . 3 ⊢ ;;655 ∈ ℕ0 |
5 | 3nn0 11909 | . . 3 ⊢ 3 ∈ ℕ0 | |
6 | 4, 5 | deccl 12107 | . 2 ⊢ ;;;6553 ∈ ℕ0 |
7 | eqid 2821 | . 2 ⊢ ;;;;65536 = ;;;;65536 | |
8 | 0nn0 11906 | . 2 ⊢ 0 ∈ ℕ0 | |
9 | 2nn0 11908 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
10 | 5, 9 | deccl 12107 | . . . . 5 ⊢ ;32 ∈ ℕ0 |
11 | 7nn0 11913 | . . . . 5 ⊢ 7 ∈ ℕ0 | |
12 | 10, 11 | deccl 12107 | . . . 4 ⊢ ;;327 ∈ ℕ0 |
13 | 12, 2 | deccl 12107 | . . 3 ⊢ ;;;3276 ∈ ℕ0 |
14 | eqid 2821 | . . . 4 ⊢ ;;;6553 = ;;;6553 | |
15 | 1nn0 11907 | . . . 4 ⊢ 1 ∈ ℕ0 | |
16 | 5p1e6 11778 | . . . . 5 ⊢ (5 + 1) = 6 | |
17 | eqid 2821 | . . . . . 6 ⊢ ;;655 = ;;655 | |
18 | eqid 2821 | . . . . . . . 8 ⊢ ;65 = ;65 | |
19 | 6t5e30 12199 | . . . . . . . . 9 ⊢ (6 · 5) = ;30 | |
20 | 2cn 11706 | . . . . . . . . . 10 ⊢ 2 ∈ ℂ | |
21 | 20 | addid2i 10822 | . . . . . . . . 9 ⊢ (0 + 2) = 2 |
22 | 5, 8, 9, 19, 21 | decaddi 12152 | . . . . . . . 8 ⊢ ((6 · 5) + 2) = ;32 |
23 | 5t5e25 12195 | . . . . . . . 8 ⊢ (5 · 5) = ;25 | |
24 | 1, 2, 1, 18, 1, 9, 22, 23 | decmul1c 12157 | . . . . . . 7 ⊢ (;65 · 5) = ;;325 |
25 | 5p2e7 11787 | . . . . . . 7 ⊢ (5 + 2) = 7 | |
26 | 10, 1, 9, 24, 25 | decaddi 12152 | . . . . . 6 ⊢ ((;65 · 5) + 2) = ;;327 |
27 | 1, 3, 1, 17, 1, 9, 26, 23 | decmul1c 12157 | . . . . 5 ⊢ (;;655 · 5) = ;;;3275 |
28 | 12, 1, 16, 27 | decsuc 12123 | . . . 4 ⊢ ((;;655 · 5) + 1) = ;;;3276 |
29 | 5cn 11719 | . . . . 5 ⊢ 5 ∈ ℂ | |
30 | 3cn 11712 | . . . . 5 ⊢ 3 ∈ ℂ | |
31 | 5t3e15 12193 | . . . . 5 ⊢ (5 · 3) = ;15 | |
32 | 29, 30, 31 | mulcomli 10644 | . . . 4 ⊢ (3 · 5) = ;15 |
33 | 1, 4, 5, 14, 1, 15, 28, 32 | decmul1c 12157 | . . 3 ⊢ (;;;6553 · 5) = ;;;;32765 |
34 | 5p3e8 11788 | . . 3 ⊢ (5 + 3) = 8 | |
35 | 13, 1, 5, 33, 34 | decaddi 12152 | . 2 ⊢ ((;;;6553 · 5) + 3) = ;;;;32768 |
36 | 1, 6, 2, 7, 8, 5, 35, 19 | decmul1c 12157 | 1 ⊢ (;;;;65536 · 5) = ;;;;;327680 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 (class class class)co 7150 0cc0 10531 1c1 10532 · cmul 10536 2c2 11686 3c3 11687 5c5 11689 6c6 11690 7c7 11691 8c8 11692 ;cdc 12092 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-ltxr 10674 df-sub 10866 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-dec 12093 |
This theorem is referenced by: fmtno5lem4 43712 |
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