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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno5faclem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for fmtno5fac 43751. (Contributed by AV, 22-Jul-2021.) |
Ref | Expression |
---|---|
fmtno5faclem2 | ⊢ (;;;;;;6700417 · 6) = ;;;;;;;40202502 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6nn0 11921 | . 2 ⊢ 6 ∈ ℕ0 | |
2 | 7nn0 11922 | . . . . . . 7 ⊢ 7 ∈ ℕ0 | |
3 | 1, 2 | deccl 12116 | . . . . . 6 ⊢ ;67 ∈ ℕ0 |
4 | 0nn0 11915 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
5 | 3, 4 | deccl 12116 | . . . . 5 ⊢ ;;670 ∈ ℕ0 |
6 | 5, 4 | deccl 12116 | . . . 4 ⊢ ;;;6700 ∈ ℕ0 |
7 | 4nn0 11919 | . . . 4 ⊢ 4 ∈ ℕ0 | |
8 | 6, 7 | deccl 12116 | . . 3 ⊢ ;;;;67004 ∈ ℕ0 |
9 | 1nn0 11916 | . . 3 ⊢ 1 ∈ ℕ0 | |
10 | 8, 9 | deccl 12116 | . 2 ⊢ ;;;;;670041 ∈ ℕ0 |
11 | eqid 2823 | . 2 ⊢ ;;;;;;6700417 = ;;;;;;6700417 | |
12 | 2nn0 11917 | . 2 ⊢ 2 ∈ ℕ0 | |
13 | 7, 4 | deccl 12116 | . . . . . . 7 ⊢ ;40 ∈ ℕ0 |
14 | 13, 12 | deccl 12116 | . . . . . 6 ⊢ ;;402 ∈ ℕ0 |
15 | 14, 4 | deccl 12116 | . . . . 5 ⊢ ;;;4020 ∈ ℕ0 |
16 | 15, 12 | deccl 12116 | . . . 4 ⊢ ;;;;40202 ∈ ℕ0 |
17 | 16, 7 | deccl 12116 | . . 3 ⊢ ;;;;;402024 ∈ ℕ0 |
18 | eqid 2823 | . . . 4 ⊢ ;;;;;670041 = ;;;;;670041 | |
19 | eqid 2823 | . . . . 5 ⊢ ;;;;67004 = ;;;;67004 | |
20 | eqid 2823 | . . . . . . 7 ⊢ ;;;6700 = ;;;6700 | |
21 | eqid 2823 | . . . . . . . 8 ⊢ ;;670 = ;;670 | |
22 | eqid 2823 | . . . . . . . . 9 ⊢ ;67 = ;67 | |
23 | 3nn0 11918 | . . . . . . . . . 10 ⊢ 3 ∈ ℕ0 | |
24 | 6t6e36 12209 | . . . . . . . . . 10 ⊢ (6 · 6) = ;36 | |
25 | 3p1e4 11785 | . . . . . . . . . 10 ⊢ (3 + 1) = 4 | |
26 | 6p4e10 12173 | . . . . . . . . . 10 ⊢ (6 + 4) = ;10 | |
27 | 23, 1, 7, 24, 25, 26 | decaddci2 12163 | . . . . . . . . 9 ⊢ ((6 · 6) + 4) = ;40 |
28 | 7t6e42 12214 | . . . . . . . . 9 ⊢ (7 · 6) = ;42 | |
29 | 1, 1, 2, 22, 12, 7, 27, 28 | decmul1c 12166 | . . . . . . . 8 ⊢ (;67 · 6) = ;;402 |
30 | 6cn 11731 | . . . . . . . . 9 ⊢ 6 ∈ ℂ | |
31 | 30 | mul02i 10831 | . . . . . . . 8 ⊢ (0 · 6) = 0 |
32 | 1, 3, 4, 21, 29, 31 | decmul1 12165 | . . . . . . 7 ⊢ (;;670 · 6) = ;;;4020 |
33 | 1, 5, 4, 20, 32, 31 | decmul1 12165 | . . . . . 6 ⊢ (;;;6700 · 6) = ;;;;40200 |
34 | 2cn 11715 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
35 | 34 | addid2i 10830 | . . . . . 6 ⊢ (0 + 2) = 2 |
36 | 15, 4, 12, 33, 35 | decaddi 12161 | . . . . 5 ⊢ ((;;;6700 · 6) + 2) = ;;;;40202 |
37 | 4cn 11725 | . . . . . 6 ⊢ 4 ∈ ℂ | |
38 | 6t4e24 12207 | . . . . . 6 ⊢ (6 · 4) = ;24 | |
39 | 30, 37, 38 | mulcomli 10652 | . . . . 5 ⊢ (4 · 6) = ;24 |
40 | 1, 6, 7, 19, 7, 12, 36, 39 | decmul1c 12166 | . . . 4 ⊢ (;;;;67004 · 6) = ;;;;;402024 |
41 | 30 | mulid2i 10648 | . . . 4 ⊢ (1 · 6) = 6 |
42 | 1, 8, 9, 18, 40, 41 | decmul1 12165 | . . 3 ⊢ (;;;;;670041 · 6) = ;;;;;;4020246 |
43 | eqid 2823 | . . . 4 ⊢ ;;;;;402024 = ;;;;;402024 | |
44 | 4p1e5 11786 | . . . 4 ⊢ (4 + 1) = 5 | |
45 | 16, 7, 9, 43, 44 | decaddi 12161 | . . 3 ⊢ (;;;;;402024 + 1) = ;;;;;402025 |
46 | 17, 1, 7, 42, 45, 26 | decaddci2 12163 | . 2 ⊢ ((;;;;;670041 · 6) + 4) = ;;;;;;4020250 |
47 | 1, 10, 2, 11, 12, 7, 46, 28 | decmul1c 12166 | 1 ⊢ (;;;;;;6700417 · 6) = ;;;;;;;40202502 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 (class class class)co 7158 0cc0 10539 1c1 10540 · cmul 10544 2c2 11695 3c3 11696 4c4 11697 5c5 11698 6c6 11699 7c7 11700 ;cdc 12101 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-ltxr 10682 df-sub 10874 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-dec 12102 |
This theorem is referenced by: fmtno5fac 43751 |
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