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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno5lem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for fmtno5 45719. (Contributed by AV, 22-Jul-2021.) |
Ref | Expression |
---|---|
fmtno5lem1 | ⊢ (;;;;65536 · 6) = ;;;;;393216 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6nn0 12431 | . 2 ⊢ 6 ∈ ℕ0 | |
2 | 5nn0 12430 | . . . . 5 ⊢ 5 ∈ ℕ0 | |
3 | 1, 2 | deccl 12630 | . . . 4 ⊢ ;65 ∈ ℕ0 |
4 | 3, 2 | deccl 12630 | . . 3 ⊢ ;;655 ∈ ℕ0 |
5 | 3nn0 12428 | . . 3 ⊢ 3 ∈ ℕ0 | |
6 | 4, 5 | deccl 12630 | . 2 ⊢ ;;;6553 ∈ ℕ0 |
7 | eqid 2736 | . 2 ⊢ ;;;;65536 = ;;;;65536 | |
8 | 9nn0 12434 | . . . . . 6 ⊢ 9 ∈ ℕ0 | |
9 | 5, 8 | deccl 12630 | . . . . 5 ⊢ ;39 ∈ ℕ0 |
10 | 9, 5 | deccl 12630 | . . . 4 ⊢ ;;393 ∈ ℕ0 |
11 | 1nn0 12426 | . . . 4 ⊢ 1 ∈ ℕ0 | |
12 | 10, 11 | deccl 12630 | . . 3 ⊢ ;;;3931 ∈ ℕ0 |
13 | 8nn0 12433 | . . 3 ⊢ 8 ∈ ℕ0 | |
14 | eqid 2736 | . . . 4 ⊢ ;;;6553 = ;;;6553 | |
15 | 0nn0 12425 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
16 | 0p1e1 12272 | . . . . 5 ⊢ (0 + 1) = 1 | |
17 | eqid 2736 | . . . . . 6 ⊢ ;;655 = ;;655 | |
18 | eqid 2736 | . . . . . . . 8 ⊢ ;65 = ;65 | |
19 | 6t6e36 12723 | . . . . . . . . 9 ⊢ (6 · 6) = ;36 | |
20 | 6p3e9 12310 | . . . . . . . . 9 ⊢ (6 + 3) = 9 | |
21 | 5, 1, 5, 19, 20 | decaddi 12675 | . . . . . . . 8 ⊢ ((6 · 6) + 3) = ;39 |
22 | 6cn 12241 | . . . . . . . . 9 ⊢ 6 ∈ ℂ | |
23 | 5cn 12238 | . . . . . . . . 9 ⊢ 5 ∈ ℂ | |
24 | 6t5e30 12722 | . . . . . . . . 9 ⊢ (6 · 5) = ;30 | |
25 | 22, 23, 24 | mulcomli 11161 | . . . . . . . 8 ⊢ (5 · 6) = ;30 |
26 | 1, 1, 2, 18, 15, 5, 21, 25 | decmul1c 12680 | . . . . . . 7 ⊢ (;65 · 6) = ;;390 |
27 | 3cn 12231 | . . . . . . . 8 ⊢ 3 ∈ ℂ | |
28 | 27 | addid2i 11340 | . . . . . . 7 ⊢ (0 + 3) = 3 |
29 | 9, 15, 5, 26, 28 | decaddi 12675 | . . . . . 6 ⊢ ((;65 · 6) + 3) = ;;393 |
30 | 1, 3, 2, 17, 15, 5, 29, 25 | decmul1c 12680 | . . . . 5 ⊢ (;;655 · 6) = ;;;3930 |
31 | 10, 15, 16, 30 | decsuc 12646 | . . . 4 ⊢ ((;;655 · 6) + 1) = ;;;3931 |
32 | 6t3e18 12720 | . . . . 5 ⊢ (6 · 3) = ;18 | |
33 | 22, 27, 32 | mulcomli 11161 | . . . 4 ⊢ (3 · 6) = ;18 |
34 | 1, 4, 5, 14, 13, 11, 31, 33 | decmul1c 12680 | . . 3 ⊢ (;;;6553 · 6) = ;;;;39318 |
35 | 1p1e2 12275 | . . . 4 ⊢ (1 + 1) = 2 | |
36 | eqid 2736 | . . . 4 ⊢ ;;;3931 = ;;;3931 | |
37 | 10, 11, 35, 36 | decsuc 12646 | . . 3 ⊢ (;;;3931 + 1) = ;;;3932 |
38 | 8p3e11 12696 | . . 3 ⊢ (8 + 3) = ;11 | |
39 | 12, 13, 5, 34, 37, 11, 38 | decaddci 12676 | . 2 ⊢ ((;;;6553 · 6) + 3) = ;;;;39321 |
40 | 1, 6, 1, 7, 1, 5, 39, 19 | decmul1c 12680 | 1 ⊢ (;;;;65536 · 6) = ;;;;;393216 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 (class class class)co 7354 0cc0 11048 1c1 11049 · cmul 11053 2c2 12205 3c3 12206 5c5 12208 6c6 12209 8c8 12211 9c9 12212 ;cdc 12615 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-addrcl 11109 ax-mulcl 11110 ax-mulrcl 11111 ax-mulcom 11112 ax-addass 11113 ax-mulass 11114 ax-distr 11115 ax-i2m1 11116 ax-1ne0 11117 ax-1rid 11118 ax-rnegex 11119 ax-rrecex 11120 ax-cnre 11121 ax-pre-lttri 11122 ax-pre-lttrn 11123 ax-pre-ltadd 11124 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7310 df-ov 7357 df-oprab 7358 df-mpo 7359 df-om 7800 df-2nd 7919 df-frecs 8209 df-wrecs 8240 df-recs 8314 df-rdg 8353 df-er 8645 df-en 8881 df-dom 8882 df-sdom 8883 df-pnf 11188 df-mnf 11189 df-ltxr 11191 df-sub 11384 df-nn 12151 df-2 12213 df-3 12214 df-4 12215 df-5 12216 df-6 12217 df-7 12218 df-8 12219 df-9 12220 df-n0 12411 df-dec 12616 |
This theorem is referenced by: fmtno5lem4 45718 |
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