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| Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno5faclem2 | Structured version Visualization version GIF version | ||
| Description: Lemma 2 for fmtno5fac 47619. (Contributed by AV, 22-Jul-2021.) |
| Ref | Expression |
|---|---|
| fmtno5faclem2 | ⊢ (;;;;;;6700417 · 6) = ;;;;;;;40202502 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 6nn0 12402 | . 2 ⊢ 6 ∈ ℕ0 | |
| 2 | 7nn0 12403 | . . . . . . 7 ⊢ 7 ∈ ℕ0 | |
| 3 | 1, 2 | deccl 12603 | . . . . . 6 ⊢ ;67 ∈ ℕ0 |
| 4 | 0nn0 12396 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
| 5 | 3, 4 | deccl 12603 | . . . . 5 ⊢ ;;670 ∈ ℕ0 |
| 6 | 5, 4 | deccl 12603 | . . . 4 ⊢ ;;;6700 ∈ ℕ0 |
| 7 | 4nn0 12400 | . . . 4 ⊢ 4 ∈ ℕ0 | |
| 8 | 6, 7 | deccl 12603 | . . 3 ⊢ ;;;;67004 ∈ ℕ0 |
| 9 | 1nn0 12397 | . . 3 ⊢ 1 ∈ ℕ0 | |
| 10 | 8, 9 | deccl 12603 | . 2 ⊢ ;;;;;670041 ∈ ℕ0 |
| 11 | eqid 2731 | . 2 ⊢ ;;;;;;6700417 = ;;;;;;6700417 | |
| 12 | 2nn0 12398 | . 2 ⊢ 2 ∈ ℕ0 | |
| 13 | 7, 4 | deccl 12603 | . . . . . . 7 ⊢ ;40 ∈ ℕ0 |
| 14 | 13, 12 | deccl 12603 | . . . . . 6 ⊢ ;;402 ∈ ℕ0 |
| 15 | 14, 4 | deccl 12603 | . . . . 5 ⊢ ;;;4020 ∈ ℕ0 |
| 16 | 15, 12 | deccl 12603 | . . . 4 ⊢ ;;;;40202 ∈ ℕ0 |
| 17 | 16, 7 | deccl 12603 | . . 3 ⊢ ;;;;;402024 ∈ ℕ0 |
| 18 | eqid 2731 | . . . 4 ⊢ ;;;;;670041 = ;;;;;670041 | |
| 19 | eqid 2731 | . . . . 5 ⊢ ;;;;67004 = ;;;;67004 | |
| 20 | eqid 2731 | . . . . . . 7 ⊢ ;;;6700 = ;;;6700 | |
| 21 | eqid 2731 | . . . . . . . 8 ⊢ ;;670 = ;;670 | |
| 22 | eqid 2731 | . . . . . . . . 9 ⊢ ;67 = ;67 | |
| 23 | 3nn0 12399 | . . . . . . . . . 10 ⊢ 3 ∈ ℕ0 | |
| 24 | 6t6e36 12696 | . . . . . . . . . 10 ⊢ (6 · 6) = ;36 | |
| 25 | 3p1e4 12265 | . . . . . . . . . 10 ⊢ (3 + 1) = 4 | |
| 26 | 6p4e10 12660 | . . . . . . . . . 10 ⊢ (6 + 4) = ;10 | |
| 27 | 23, 1, 7, 24, 25, 26 | decaddci2 12650 | . . . . . . . . 9 ⊢ ((6 · 6) + 4) = ;40 |
| 28 | 7t6e42 12701 | . . . . . . . . 9 ⊢ (7 · 6) = ;42 | |
| 29 | 1, 1, 2, 22, 12, 7, 27, 28 | decmul1c 12653 | . . . . . . . 8 ⊢ (;67 · 6) = ;;402 |
| 30 | 6cn 12216 | . . . . . . . . 9 ⊢ 6 ∈ ℂ | |
| 31 | 30 | mul02i 11302 | . . . . . . . 8 ⊢ (0 · 6) = 0 |
| 32 | 1, 3, 4, 21, 29, 31 | decmul1 12652 | . . . . . . 7 ⊢ (;;670 · 6) = ;;;4020 |
| 33 | 1, 5, 4, 20, 32, 31 | decmul1 12652 | . . . . . 6 ⊢ (;;;6700 · 6) = ;;;;40200 |
| 34 | 2cn 12200 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
| 35 | 34 | addlidi 11301 | . . . . . 6 ⊢ (0 + 2) = 2 |
| 36 | 15, 4, 12, 33, 35 | decaddi 12648 | . . . . 5 ⊢ ((;;;6700 · 6) + 2) = ;;;;40202 |
| 37 | 4cn 12210 | . . . . . 6 ⊢ 4 ∈ ℂ | |
| 38 | 6t4e24 12694 | . . . . . 6 ⊢ (6 · 4) = ;24 | |
| 39 | 30, 37, 38 | mulcomli 11121 | . . . . 5 ⊢ (4 · 6) = ;24 |
| 40 | 1, 6, 7, 19, 7, 12, 36, 39 | decmul1c 12653 | . . . 4 ⊢ (;;;;67004 · 6) = ;;;;;402024 |
| 41 | 30 | mullidi 11117 | . . . 4 ⊢ (1 · 6) = 6 |
| 42 | 1, 8, 9, 18, 40, 41 | decmul1 12652 | . . 3 ⊢ (;;;;;670041 · 6) = ;;;;;;4020246 |
| 43 | eqid 2731 | . . . 4 ⊢ ;;;;;402024 = ;;;;;402024 | |
| 44 | 4p1e5 12266 | . . . 4 ⊢ (4 + 1) = 5 | |
| 45 | 16, 7, 9, 43, 44 | decaddi 12648 | . . 3 ⊢ (;;;;;402024 + 1) = ;;;;;402025 |
| 46 | 17, 1, 7, 42, 45, 26 | decaddci2 12650 | . 2 ⊢ ((;;;;;670041 · 6) + 4) = ;;;;;;4020250 |
| 47 | 1, 10, 2, 11, 12, 7, 46, 28 | decmul1c 12653 | 1 ⊢ (;;;;;;6700417 · 6) = ;;;;;;;40202502 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1541 (class class class)co 7346 0cc0 11006 1c1 11007 · cmul 11011 2c2 12180 3c3 12181 4c4 12182 5c5 12183 6c6 12184 7c7 12185 ;cdc 12588 |
| 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 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-pnf 11148 df-mnf 11149 df-ltxr 11151 df-sub 11346 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-dec 12589 |
| This theorem is referenced by: fmtno5fac 47619 |
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