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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno5lem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for fmtno5 40007. (Contributed by AV, 22-Jul-2021.) |
Ref | Expression |
---|---|
fmtno5lem1 | ⊢ (;;;;65536 · 6) = ;;;;;393216 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6nn0 11190 | . 2 ⊢ 6 ∈ ℕ0 | |
2 | 5nn0 11189 | . . . . 5 ⊢ 5 ∈ ℕ0 | |
3 | 1, 2 | deccl 11388 | . . . 4 ⊢ ;65 ∈ ℕ0 |
4 | 3, 2 | deccl 11388 | . . 3 ⊢ ;;655 ∈ ℕ0 |
5 | 3nn0 11187 | . . 3 ⊢ 3 ∈ ℕ0 | |
6 | 4, 5 | deccl 11388 | . 2 ⊢ ;;;6553 ∈ ℕ0 |
7 | eqid 2610 | . 2 ⊢ ;;;;65536 = ;;;;65536 | |
8 | 9nn0 11193 | . . . . . 6 ⊢ 9 ∈ ℕ0 | |
9 | 5, 8 | deccl 11388 | . . . . 5 ⊢ ;39 ∈ ℕ0 |
10 | 9, 5 | deccl 11388 | . . . 4 ⊢ ;;393 ∈ ℕ0 |
11 | 1nn0 11185 | . . . 4 ⊢ 1 ∈ ℕ0 | |
12 | 10, 11 | deccl 11388 | . . 3 ⊢ ;;;3931 ∈ ℕ0 |
13 | 8nn0 11192 | . . 3 ⊢ 8 ∈ ℕ0 | |
14 | eqid 2610 | . . . 4 ⊢ ;;;6553 = ;;;6553 | |
15 | 0nn0 11184 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
16 | 0p1e1 11009 | . . . . 5 ⊢ (0 + 1) = 1 | |
17 | eqid 2610 | . . . . . 6 ⊢ ;;655 = ;;655 | |
18 | eqid 2610 | . . . . . . . 8 ⊢ ;65 = ;65 | |
19 | 6t6e36 11522 | . . . . . . . . 9 ⊢ (6 · 6) = ;36 | |
20 | 6p3e9 11047 | . . . . . . . . 9 ⊢ (6 + 3) = 9 | |
21 | 5, 1, 5, 19, 20 | decaddi 11455 | . . . . . . . 8 ⊢ ((6 · 6) + 3) = ;39 |
22 | 6cn 10979 | . . . . . . . . 9 ⊢ 6 ∈ ℂ | |
23 | 5cn 10977 | . . . . . . . . 9 ⊢ 5 ∈ ℂ | |
24 | 6t5e30 11520 | . . . . . . . . 9 ⊢ (6 · 5) = ;30 | |
25 | 22, 23, 24 | mulcomli 9926 | . . . . . . . 8 ⊢ (5 · 6) = ;30 |
26 | 1, 1, 2, 18, 15, 5, 21, 25 | decmul1c 11463 | . . . . . . 7 ⊢ (;65 · 6) = ;;390 |
27 | 3cn 10972 | . . . . . . . 8 ⊢ 3 ∈ ℂ | |
28 | 27 | addid2i 10103 | . . . . . . 7 ⊢ (0 + 3) = 3 |
29 | 9, 15, 5, 26, 28 | decaddi 11455 | . . . . . 6 ⊢ ((;65 · 6) + 3) = ;;393 |
30 | 1, 3, 2, 17, 15, 5, 29, 25 | decmul1c 11463 | . . . . 5 ⊢ (;;655 · 6) = ;;;3930 |
31 | 10, 15, 16, 30 | decsuc 11411 | . . . 4 ⊢ ((;;655 · 6) + 1) = ;;;3931 |
32 | 6t3e18 11518 | . . . . 5 ⊢ (6 · 3) = ;18 | |
33 | 22, 27, 32 | mulcomli 9926 | . . . 4 ⊢ (3 · 6) = ;18 |
34 | 1, 4, 5, 14, 13, 11, 31, 33 | decmul1c 11463 | . . 3 ⊢ (;;;6553 · 6) = ;;;;39318 |
35 | 1p1e2 11011 | . . . 4 ⊢ (1 + 1) = 2 | |
36 | eqid 2610 | . . . 4 ⊢ ;;;3931 = ;;;3931 | |
37 | 10, 11, 35, 36 | decsuc 11411 | . . 3 ⊢ (;;;3931 + 1) = ;;;3932 |
38 | 8p3e11 11488 | . . 3 ⊢ (8 + 3) = ;11 | |
39 | 12, 13, 5, 34, 37, 11, 38 | decaddci 11456 | . 2 ⊢ ((;;;6553 · 6) + 3) = ;;;;39321 |
40 | 1, 6, 1, 7, 1, 5, 39, 19 | decmul1c 11463 | 1 ⊢ (;;;;65536 · 6) = ;;;;;393216 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 (class class class)co 6549 0cc0 9815 1c1 9816 · cmul 9820 2c2 10947 3c3 10948 5c5 10950 6c6 10951 8c8 10953 9c9 10954 ;cdc 11369 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-sub 10147 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-dec 11370 |
This theorem is referenced by: fmtno5lem4 40006 |
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