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Mirrors > Home > MPE Home > Th. List > xmullem | Unicode version |
Description: Lemma for rexmul 11492. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xmullem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ioran 490 | . . . 4 | |
2 | 1 | anbi2i 694 | . . 3 |
3 | ioran 490 | . . . . 5 | |
4 | ioran 490 | . . . . . 6 | |
5 | ioran 490 | . . . . . 6 | |
6 | 4, 5 | anbi12i 697 | . . . . 5 |
7 | 3, 6 | bitri 249 | . . . 4 |
8 | ioran 490 | . . . . 5 | |
9 | ioran 490 | . . . . . 6 | |
10 | ioran 490 | . . . . . 6 | |
11 | 9, 10 | anbi12i 697 | . . . . 5 |
12 | 8, 11 | bitri 249 | . . . 4 |
13 | 7, 12 | anbi12i 697 | . . 3 |
14 | simplll 759 | . . . . 5 | |
15 | elxr 11354 | . . . . 5 | |
16 | 14, 15 | sylib 196 | . . . 4 |
17 | idd 24 | . . . . 5 | |
18 | simprlr 764 | . . . . . . . . 9 | |
19 | 18 | adantl 466 | . . . . . . . 8 |
20 | 19 | pm2.21d 106 | . . . . . . 7 |
21 | 20 | expdimp 437 | . . . . . 6 |
22 | simplrr 762 | . . . . . . . 8 | |
23 | 22 | pm2.21d 106 | . . . . . . 7 |
24 | 23 | imp 429 | . . . . . 6 |
25 | simplll 759 | . . . . . . . . 9 | |
26 | 25 | adantl 466 | . . . . . . . 8 |
27 | 26 | pm2.21d 106 | . . . . . . 7 |
28 | 27 | expdimp 437 | . . . . . 6 |
29 | simpllr 760 | . . . . . . 7 | |
30 | 0xr 9661 | . . . . . . 7 | |
31 | xrltso 11376 | . . . . . . . 8 | |
32 | solin 4828 | . . . . . . . 8 | |
33 | 31, 32 | mpan 670 | . . . . . . 7 |
34 | 29, 30, 33 | sylancl 662 | . . . . . 6 |
35 | 21, 24, 28, 34 | mpjao3dan 1295 | . . . . 5 |
36 | simpllr 760 | . . . . . . . . 9 | |
37 | 36 | adantl 466 | . . . . . . . 8 |
38 | 37 | pm2.21d 106 | . . . . . . 7 |
39 | 38 | expdimp 437 | . . . . . 6 |
40 | 22 | pm2.21d 106 | . . . . . . 7 |
41 | 40 | imp 429 | . . . . . 6 |
42 | simprll 763 | . . . . . . . . 9 | |
43 | 42 | adantl 466 | . . . . . . . 8 |
44 | 43 | pm2.21d 106 | . . . . . . 7 |
45 | 44 | expdimp 437 | . . . . . 6 |
46 | 39, 41, 45, 34 | mpjao3dan 1295 | . . . . 5 |
47 | 17, 35, 46 | 3jaod 1292 | . . . 4 |
48 | 16, 47 | mpd 15 | . . 3 |
49 | 2, 13, 48 | syl2anb 479 | . 2 |
50 | 49 | anassrs 648 | 1 |
Colors of variables: wff setvar class |
Syntax hints: -. wn 3 -> wi 4
\/ wo 368 /\ wa 369 \/ w3o 972
= wceq 1395 e. wcel 1818 class class class wbr 4452
Or wor 4804 cr 9512 0 cc0 9513 cpnf 9646 cmnf 9647
cxr 9648
clt 9649 |
This theorem is referenced by: xmulcom 11487 xmulneg1 11490 xmulf 11493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-8 1820 ax-9 1822 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 ax-sep 4573 ax-nul 4581 ax-pow 4630 ax-pr 4691 ax-un 6592 ax-cnex 9569 ax-resscn 9570 ax-1cn 9571 ax-icn 9572 ax-addcl 9573 ax-addrcl 9574 ax-mulcl 9575 ax-mulrcl 9576 ax-i2m1 9581 ax-1ne0 9582 ax-rnegex 9584 ax-rrecex 9585 ax-cnre 9586 ax-pre-lttri 9587 ax-pre-lttrn 9588 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 974 df-3an 975 df-tru 1398 df-ex 1613 df-nf 1617 df-sb 1740 df-eu 2286 df-mo 2287 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ne 2654 df-nel 2655 df-ral 2812 df-rex 2813 df-rab 2816 df-v 3111 df-sbc 3328 df-csb 3435 df-dif 3478 df-un 3480 df-in 3482 df-ss 3489 df-nul 3785 df-if 3942 df-pw 4014 df-sn 4030 df-pr 4032 df-op 4036 df-uni 4250 df-br 4453 df-opab 4511 df-mpt 4512 df-id 4800 df-po 4805 df-so 4806 df-xp 5010 df-rel 5011 df-cnv 5012 df-co 5013 df-dm 5014 df-rn 5015 df-res 5016 df-ima 5017 df-iota 5556 df-fun 5595 df-fn 5596 df-f 5597 df-f1 5598 df-fo 5599 df-f1o 5600 df-fv 5601 df-ov 6299 df-er 7330 df-en 7537 df-dom 7538 df-sdom 7539 df-pnf 9651 df-mnf 9652 df-xr 9653 df-ltxr 9654 |
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