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Mirrors > Home > MPE Home > Th. List > 4t3lem | Structured version Visualization version GIF version |
Description: Lemma for 4t3e12 12723 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
4t3lem.1 | โข ๐ด โ โ0 |
4t3lem.2 | โข ๐ต โ โ0 |
4t3lem.3 | โข ๐ถ = (๐ต + 1) |
4t3lem.4 | โข (๐ด ยท ๐ต) = ๐ท |
4t3lem.5 | โข (๐ท + ๐ด) = ๐ธ |
Ref | Expression |
---|---|
4t3lem | โข (๐ด ยท ๐ถ) = ๐ธ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4t3lem.3 | . . 3 โข ๐ถ = (๐ต + 1) | |
2 | 1 | oveq2i 7373 | . 2 โข (๐ด ยท ๐ถ) = (๐ด ยท (๐ต + 1)) |
3 | 4t3lem.1 | . . . . . 6 โข ๐ด โ โ0 | |
4 | 3 | nn0cni 12432 | . . . . 5 โข ๐ด โ โ |
5 | 4t3lem.2 | . . . . . 6 โข ๐ต โ โ0 | |
6 | 5 | nn0cni 12432 | . . . . 5 โข ๐ต โ โ |
7 | ax-1cn 11116 | . . . . 5 โข 1 โ โ | |
8 | 4, 6, 7 | adddii 11174 | . . . 4 โข (๐ด ยท (๐ต + 1)) = ((๐ด ยท ๐ต) + (๐ด ยท 1)) |
9 | 4t3lem.4 | . . . . 5 โข (๐ด ยท ๐ต) = ๐ท | |
10 | 4 | mulid1i 11166 | . . . . 5 โข (๐ด ยท 1) = ๐ด |
11 | 9, 10 | oveq12i 7374 | . . . 4 โข ((๐ด ยท ๐ต) + (๐ด ยท 1)) = (๐ท + ๐ด) |
12 | 8, 11 | eqtri 2765 | . . 3 โข (๐ด ยท (๐ต + 1)) = (๐ท + ๐ด) |
13 | 4t3lem.5 | . . 3 โข (๐ท + ๐ด) = ๐ธ | |
14 | 12, 13 | eqtri 2765 | . 2 โข (๐ด ยท (๐ต + 1)) = ๐ธ |
15 | 2, 14 | eqtri 2765 | 1 โข (๐ด ยท ๐ถ) = ๐ธ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 โ wcel 2107 (class class class)co 7362 1c1 11059 + caddc 11061 ยท cmul 11063 โ0cn0 12420 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-un 7677 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-mulcl 11120 ax-mulcom 11122 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1rid 11128 ax-cnre 11131 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-nn 12161 df-n0 12421 |
This theorem is referenced by: 4t3e12 12723 4t4e16 12724 5t2e10 12725 5t3e15 12726 5t4e20 12727 5t5e25 12728 6t3e18 12730 6t4e24 12731 6t5e30 12732 6t6e36 12733 7t3e21 12735 7t4e28 12736 7t5e35 12737 7t6e42 12738 7t7e49 12739 8t3e24 12741 8t4e32 12742 8t5e40 12743 8t6e48 12744 8t7e56 12745 8t8e64 12746 9t3e27 12748 9t4e36 12749 9t5e45 12750 9t6e54 12751 9t7e63 12752 9t8e72 12753 9t9e81 12754 |
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