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Mirrors > Home > MPE Home > Th. List > fin1a2lem3 | Structured version Visualization version GIF version |
Description: Lemma for fin1a2 10412. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
Ref | Expression |
---|---|
fin1a2lem.b | โข ๐ธ = (๐ฅ โ ฯ โฆ (2o ยทo ๐ฅ)) |
Ref | Expression |
---|---|
fin1a2lem3 | โข (๐ด โ ฯ โ (๐ธโ๐ด) = (2o ยทo ๐ด)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7419 | . 2 โข (๐ = ๐ด โ (2o ยทo ๐) = (2o ยทo ๐ด)) | |
2 | fin1a2lem.b | . . 3 โข ๐ธ = (๐ฅ โ ฯ โฆ (2o ยทo ๐ฅ)) | |
3 | oveq2 7419 | . . . 4 โข (๐ฅ = ๐ โ (2o ยทo ๐ฅ) = (2o ยทo ๐)) | |
4 | 3 | cbvmptv 5261 | . . 3 โข (๐ฅ โ ฯ โฆ (2o ยทo ๐ฅ)) = (๐ โ ฯ โฆ (2o ยทo ๐)) |
5 | 2, 4 | eqtri 2760 | . 2 โข ๐ธ = (๐ โ ฯ โฆ (2o ยทo ๐)) |
6 | ovex 7444 | . 2 โข (2o ยทo ๐ด) โ V | |
7 | 1, 5, 6 | fvmpt 6998 | 1 โข (๐ด โ ฯ โ (๐ธโ๐ด) = (2o ยทo ๐ด)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1541 โ wcel 2106 โฆ cmpt 5231 โcfv 6543 (class class class)co 7411 ฯcom 7857 2oc2o 8462 ยทo comu 8466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7414 |
This theorem is referenced by: fin1a2lem4 10400 fin1a2lem5 10401 fin1a2lem6 10402 |
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