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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno5faclem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for fmtno5fac 42519. (Contributed by AV, 22-Jul-2021.) |
Ref | Expression |
---|---|
fmtno5faclem2 | ⊢ (;;;;;;6700417 · 6) = ;;;;;;;40202502 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6nn0 11665 | . 2 ⊢ 6 ∈ ℕ0 | |
2 | 7nn0 11666 | . . . . . . 7 ⊢ 7 ∈ ℕ0 | |
3 | 1, 2 | deccl 11860 | . . . . . 6 ⊢ ;67 ∈ ℕ0 |
4 | 0nn0 11659 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
5 | 3, 4 | deccl 11860 | . . . . 5 ⊢ ;;670 ∈ ℕ0 |
6 | 5, 4 | deccl 11860 | . . . 4 ⊢ ;;;6700 ∈ ℕ0 |
7 | 4nn0 11663 | . . . 4 ⊢ 4 ∈ ℕ0 | |
8 | 6, 7 | deccl 11860 | . . 3 ⊢ ;;;;67004 ∈ ℕ0 |
9 | 1nn0 11660 | . . 3 ⊢ 1 ∈ ℕ0 | |
10 | 8, 9 | deccl 11860 | . 2 ⊢ ;;;;;670041 ∈ ℕ0 |
11 | eqid 2778 | . 2 ⊢ ;;;;;;6700417 = ;;;;;;6700417 | |
12 | 2nn0 11661 | . 2 ⊢ 2 ∈ ℕ0 | |
13 | 7, 4 | deccl 11860 | . . . . . . 7 ⊢ ;40 ∈ ℕ0 |
14 | 13, 12 | deccl 11860 | . . . . . 6 ⊢ ;;402 ∈ ℕ0 |
15 | 14, 4 | deccl 11860 | . . . . 5 ⊢ ;;;4020 ∈ ℕ0 |
16 | 15, 12 | deccl 11860 | . . . 4 ⊢ ;;;;40202 ∈ ℕ0 |
17 | 16, 7 | deccl 11860 | . . 3 ⊢ ;;;;;402024 ∈ ℕ0 |
18 | eqid 2778 | . . . 4 ⊢ ;;;;;670041 = ;;;;;670041 | |
19 | eqid 2778 | . . . . 5 ⊢ ;;;;67004 = ;;;;67004 | |
20 | eqid 2778 | . . . . . . 7 ⊢ ;;;6700 = ;;;6700 | |
21 | eqid 2778 | . . . . . . . 8 ⊢ ;;670 = ;;670 | |
22 | eqid 2778 | . . . . . . . . 9 ⊢ ;67 = ;67 | |
23 | 3nn0 11662 | . . . . . . . . . 10 ⊢ 3 ∈ ℕ0 | |
24 | 6t6e36 11955 | . . . . . . . . . 10 ⊢ (6 · 6) = ;36 | |
25 | 3p1e4 11527 | . . . . . . . . . 10 ⊢ (3 + 1) = 4 | |
26 | 6p4e10 11919 | . . . . . . . . . 10 ⊢ (6 + 4) = ;10 | |
27 | 23, 1, 7, 24, 25, 26 | decaddci2 11908 | . . . . . . . . 9 ⊢ ((6 · 6) + 4) = ;40 |
28 | 7t6e42 11960 | . . . . . . . . 9 ⊢ (7 · 6) = ;42 | |
29 | 1, 1, 2, 22, 12, 7, 27, 28 | decmul1c 11912 | . . . . . . . 8 ⊢ (;67 · 6) = ;;402 |
30 | 6cn 11469 | . . . . . . . . 9 ⊢ 6 ∈ ℂ | |
31 | 30 | mul02i 10565 | . . . . . . . 8 ⊢ (0 · 6) = 0 |
32 | 1, 3, 4, 21, 29, 31 | decmul1 11910 | . . . . . . 7 ⊢ (;;670 · 6) = ;;;4020 |
33 | 1, 5, 4, 20, 32, 31 | decmul1 11910 | . . . . . 6 ⊢ (;;;6700 · 6) = ;;;;40200 |
34 | 2cn 11450 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
35 | 34 | addid2i 10564 | . . . . . 6 ⊢ (0 + 2) = 2 |
36 | 15, 4, 12, 33, 35 | decaddi 11906 | . . . . 5 ⊢ ((;;;6700 · 6) + 2) = ;;;;40202 |
37 | 4cn 11461 | . . . . . 6 ⊢ 4 ∈ ℂ | |
38 | 6t4e24 11953 | . . . . . 6 ⊢ (6 · 4) = ;24 | |
39 | 30, 37, 38 | mulcomli 10386 | . . . . 5 ⊢ (4 · 6) = ;24 |
40 | 1, 6, 7, 19, 7, 12, 36, 39 | decmul1c 11912 | . . . 4 ⊢ (;;;;67004 · 6) = ;;;;;402024 |
41 | 30 | mulid2i 10382 | . . . 4 ⊢ (1 · 6) = 6 |
42 | 1, 8, 9, 18, 40, 41 | decmul1 11910 | . . 3 ⊢ (;;;;;670041 · 6) = ;;;;;;4020246 |
43 | eqid 2778 | . . . 4 ⊢ ;;;;;402024 = ;;;;;402024 | |
44 | 4p1e5 11528 | . . . 4 ⊢ (4 + 1) = 5 | |
45 | 16, 7, 9, 43, 44 | decaddi 11906 | . . 3 ⊢ (;;;;;402024 + 1) = ;;;;;402025 |
46 | 17, 1, 7, 42, 45, 26 | decaddci2 11908 | . 2 ⊢ ((;;;;;670041 · 6) + 4) = ;;;;;;4020250 |
47 | 1, 10, 2, 11, 12, 7, 46, 28 | decmul1c 11912 | 1 ⊢ (;;;;;;6700417 · 6) = ;;;;;;;40202502 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 (class class class)co 6922 0cc0 10272 1c1 10273 · cmul 10277 2c2 11430 3c3 11431 4c4 11432 5c5 11433 6c6 11434 7c7 11435 ;cdc 11845 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-ltxr 10416 df-sub 10608 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-dec 11846 |
This theorem is referenced by: fmtno5fac 42519 |
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