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Mirrors > Home > MPE Home > Th. List > 4t3lem | Structured version Visualization version GIF version |
Description: Lemma for 4t3e12 12775 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 7420 | . 2 โข (๐ด ยท ๐ถ) = (๐ด ยท (๐ต + 1)) |
3 | 4t3lem.1 | . . . . . 6 โข ๐ด โ โ0 | |
4 | 3 | nn0cni 12484 | . . . . 5 โข ๐ด โ โ |
5 | 4t3lem.2 | . . . . . 6 โข ๐ต โ โ0 | |
6 | 5 | nn0cni 12484 | . . . . 5 โข ๐ต โ โ |
7 | ax-1cn 11168 | . . . . 5 โข 1 โ โ | |
8 | 4, 6, 7 | adddii 11226 | . . . 4 โข (๐ด ยท (๐ต + 1)) = ((๐ด ยท ๐ต) + (๐ด ยท 1)) |
9 | 4t3lem.4 | . . . . 5 โข (๐ด ยท ๐ต) = ๐ท | |
10 | 4 | mulridi 11218 | . . . . 5 โข (๐ด ยท 1) = ๐ด |
11 | 9, 10 | oveq12i 7421 | . . . 4 โข ((๐ด ยท ๐ต) + (๐ด ยท 1)) = (๐ท + ๐ด) |
12 | 8, 11 | eqtri 2761 | . . 3 โข (๐ด ยท (๐ต + 1)) = (๐ท + ๐ด) |
13 | 4t3lem.5 | . . 3 โข (๐ท + ๐ด) = ๐ธ | |
14 | 12, 13 | eqtri 2761 | . 2 โข (๐ด ยท (๐ต + 1)) = ๐ธ |
15 | 2, 14 | eqtri 2761 | 1 โข (๐ด ยท ๐ถ) = ๐ธ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 โ wcel 2107 (class class class)co 7409 1c1 11111 + caddc 11113 ยท cmul 11115 โ0cn0 12472 |
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 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-mulcl 11172 ax-mulcom 11174 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1rid 11180 ax-cnre 11183 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-nn 12213 df-n0 12473 |
This theorem is referenced by: 4t3e12 12775 4t4e16 12776 5t2e10 12777 5t3e15 12778 5t4e20 12779 5t5e25 12780 6t3e18 12782 6t4e24 12783 6t5e30 12784 6t6e36 12785 7t3e21 12787 7t4e28 12788 7t5e35 12789 7t6e42 12790 7t7e49 12791 8t3e24 12793 8t4e32 12794 8t5e40 12795 8t6e48 12796 8t7e56 12797 8t8e64 12798 9t3e27 12800 9t4e36 12801 9t5e45 12802 9t6e54 12803 9t7e63 12804 9t8e72 12805 9t9e81 12806 |
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