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| Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno5faclem2 | Structured version Visualization version GIF version | ||
| Description: Lemma 2 for fmtno5fac 47569. (Contributed by AV, 22-Jul-2021.) | 
| Ref | Expression | 
|---|---|
| fmtno5faclem2 | ⊢ (;;;;;;6700417 · 6) = ;;;;;;;40202502 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 6nn0 12547 | . 2 ⊢ 6 ∈ ℕ0 | |
| 2 | 7nn0 12548 | . . . . . . 7 ⊢ 7 ∈ ℕ0 | |
| 3 | 1, 2 | deccl 12748 | . . . . . 6 ⊢ ;67 ∈ ℕ0 | 
| 4 | 0nn0 12541 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
| 5 | 3, 4 | deccl 12748 | . . . . 5 ⊢ ;;670 ∈ ℕ0 | 
| 6 | 5, 4 | deccl 12748 | . . . 4 ⊢ ;;;6700 ∈ ℕ0 | 
| 7 | 4nn0 12545 | . . . 4 ⊢ 4 ∈ ℕ0 | |
| 8 | 6, 7 | deccl 12748 | . . 3 ⊢ ;;;;67004 ∈ ℕ0 | 
| 9 | 1nn0 12542 | . . 3 ⊢ 1 ∈ ℕ0 | |
| 10 | 8, 9 | deccl 12748 | . 2 ⊢ ;;;;;670041 ∈ ℕ0 | 
| 11 | eqid 2737 | . 2 ⊢ ;;;;;;6700417 = ;;;;;;6700417 | |
| 12 | 2nn0 12543 | . 2 ⊢ 2 ∈ ℕ0 | |
| 13 | 7, 4 | deccl 12748 | . . . . . . 7 ⊢ ;40 ∈ ℕ0 | 
| 14 | 13, 12 | deccl 12748 | . . . . . 6 ⊢ ;;402 ∈ ℕ0 | 
| 15 | 14, 4 | deccl 12748 | . . . . 5 ⊢ ;;;4020 ∈ ℕ0 | 
| 16 | 15, 12 | deccl 12748 | . . . 4 ⊢ ;;;;40202 ∈ ℕ0 | 
| 17 | 16, 7 | deccl 12748 | . . 3 ⊢ ;;;;;402024 ∈ ℕ0 | 
| 18 | eqid 2737 | . . . 4 ⊢ ;;;;;670041 = ;;;;;670041 | |
| 19 | eqid 2737 | . . . . 5 ⊢ ;;;;67004 = ;;;;67004 | |
| 20 | eqid 2737 | . . . . . . 7 ⊢ ;;;6700 = ;;;6700 | |
| 21 | eqid 2737 | . . . . . . . 8 ⊢ ;;670 = ;;670 | |
| 22 | eqid 2737 | . . . . . . . . 9 ⊢ ;67 = ;67 | |
| 23 | 3nn0 12544 | . . . . . . . . . 10 ⊢ 3 ∈ ℕ0 | |
| 24 | 6t6e36 12841 | . . . . . . . . . 10 ⊢ (6 · 6) = ;36 | |
| 25 | 3p1e4 12411 | . . . . . . . . . 10 ⊢ (3 + 1) = 4 | |
| 26 | 6p4e10 12805 | . . . . . . . . . 10 ⊢ (6 + 4) = ;10 | |
| 27 | 23, 1, 7, 24, 25, 26 | decaddci2 12795 | . . . . . . . . 9 ⊢ ((6 · 6) + 4) = ;40 | 
| 28 | 7t6e42 12846 | . . . . . . . . 9 ⊢ (7 · 6) = ;42 | |
| 29 | 1, 1, 2, 22, 12, 7, 27, 28 | decmul1c 12798 | . . . . . . . 8 ⊢ (;67 · 6) = ;;402 | 
| 30 | 6cn 12357 | . . . . . . . . 9 ⊢ 6 ∈ ℂ | |
| 31 | 30 | mul02i 11450 | . . . . . . . 8 ⊢ (0 · 6) = 0 | 
| 32 | 1, 3, 4, 21, 29, 31 | decmul1 12797 | . . . . . . 7 ⊢ (;;670 · 6) = ;;;4020 | 
| 33 | 1, 5, 4, 20, 32, 31 | decmul1 12797 | . . . . . 6 ⊢ (;;;6700 · 6) = ;;;;40200 | 
| 34 | 2cn 12341 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
| 35 | 34 | addlidi 11449 | . . . . . 6 ⊢ (0 + 2) = 2 | 
| 36 | 15, 4, 12, 33, 35 | decaddi 12793 | . . . . 5 ⊢ ((;;;6700 · 6) + 2) = ;;;;40202 | 
| 37 | 4cn 12351 | . . . . . 6 ⊢ 4 ∈ ℂ | |
| 38 | 6t4e24 12839 | . . . . . 6 ⊢ (6 · 4) = ;24 | |
| 39 | 30, 37, 38 | mulcomli 11270 | . . . . 5 ⊢ (4 · 6) = ;24 | 
| 40 | 1, 6, 7, 19, 7, 12, 36, 39 | decmul1c 12798 | . . . 4 ⊢ (;;;;67004 · 6) = ;;;;;402024 | 
| 41 | 30 | mullidi 11266 | . . . 4 ⊢ (1 · 6) = 6 | 
| 42 | 1, 8, 9, 18, 40, 41 | decmul1 12797 | . . 3 ⊢ (;;;;;670041 · 6) = ;;;;;;4020246 | 
| 43 | eqid 2737 | . . . 4 ⊢ ;;;;;402024 = ;;;;;402024 | |
| 44 | 4p1e5 12412 | . . . 4 ⊢ (4 + 1) = 5 | |
| 45 | 16, 7, 9, 43, 44 | decaddi 12793 | . . 3 ⊢ (;;;;;402024 + 1) = ;;;;;402025 | 
| 46 | 17, 1, 7, 42, 45, 26 | decaddci2 12795 | . 2 ⊢ ((;;;;;670041 · 6) + 4) = ;;;;;;4020250 | 
| 47 | 1, 10, 2, 11, 12, 7, 46, 28 | decmul1c 12798 | 1 ⊢ (;;;;;;6700417 · 6) = ;;;;;;;40202502 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 (class class class)co 7431 0cc0 11155 1c1 11156 · cmul 11160 2c2 12321 3c3 12322 4c4 12323 5c5 12324 6c6 12325 7c7 12326 ;cdc 12733 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-sub 11494 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-dec 12734 | 
| This theorem is referenced by: fmtno5fac 47569 | 
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