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Mirrors > Home > ILE Home > Th. List > prarloclem5 | Unicode version |
Description: A substitution of zero for and minus two for . Lemma for prarloc 6693. (Contributed by Jim Kingdon, 4-Nov-2019.) |
Ref | Expression |
---|---|
prarloclem5 | +Q0 ~Q0 ·Q0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prarloclemn 6689 | . . . 4 | |
2 | 1 | 3adant2 957 | . . 3 |
3 | 2 | 3ad2ant2 960 | . 2 |
4 | elprnql 6671 | . . . . . . 7 | |
5 | 4 | 3ad2ant1 959 | . . . . . 6 |
6 | simp22 972 | . . . . . 6 | |
7 | nqnq0 6631 | . . . . . . . . 9 Q0 | |
8 | 7 | sseli 2995 | . . . . . . . 8 Q0 |
9 | nq0a0 6647 | . . . . . . . 8 Q0 +Q0 0Q0 | |
10 | 8, 9 | syl 14 | . . . . . . 7 +Q0 0Q0 |
11 | df-0nq0 6616 | . . . . . . . . . 10 0Q0 ~Q0 | |
12 | 11 | oveq1i 5542 | . . . . . . . . 9 0Q0 ·Q0 ~Q0 ·Q0 |
13 | 7 | sseli 2995 | . . . . . . . . . 10 Q0 |
14 | nq0m0r 6646 | . . . . . . . . . 10 Q0 0Q0 ·Q0 0Q0 | |
15 | 13, 14 | syl 14 | . . . . . . . . 9 0Q0 ·Q0 0Q0 |
16 | 12, 15 | syl5reqr 2128 | . . . . . . . 8 0Q0 ~Q0 ·Q0 |
17 | 16 | oveq2d 5548 | . . . . . . 7 +Q0 0Q0 +Q0 ~Q0 ·Q0 |
18 | 10, 17 | sylan9req 2134 | . . . . . 6 +Q0 ~Q0 ·Q0 |
19 | 5, 6, 18 | syl2anc 403 | . . . . 5 +Q0 ~Q0 ·Q0 |
20 | simp1r 963 | . . . . 5 | |
21 | 19, 20 | eqeltrrd 2156 | . . . 4 +Q0 ~Q0 ·Q0 |
22 | 2onn 6117 | . . . . . . . . . . . . . . 15 | |
23 | nna0r 6080 | . . . . . . . . . . . . . . 15 | |
24 | 22, 23 | ax-mp 7 | . . . . . . . . . . . . . 14 |
25 | 24 | oveq1i 5542 | . . . . . . . . . . . . 13 |
26 | 25 | eqeq1i 2088 | . . . . . . . . . . . 12 |
27 | 26 | biimpri 131 | . . . . . . . . . . 11 |
28 | 27 | opeq1d 3576 | . . . . . . . . . 10 |
29 | 28 | eceq1d 6165 | . . . . . . . . 9 |
30 | 29 | oveq1d 5547 | . . . . . . . 8 |
31 | 30 | oveq2d 5548 | . . . . . . 7 |
32 | 31 | eleq1d 2147 | . . . . . 6 |
33 | 32 | biimprcd 158 | . . . . 5 |
34 | 33 | 3ad2ant3 961 | . . . 4 |
35 | peano1 4335 | . . . . 5 | |
36 | opeq1 3570 | . . . . . . . . . . 11 | |
37 | 36 | eceq1d 6165 | . . . . . . . . . 10 ~Q0 ~Q0 |
38 | 37 | oveq1d 5547 | . . . . . . . . 9 ~Q0 ·Q0 ~Q0 ·Q0 |
39 | 38 | oveq2d 5548 | . . . . . . . 8 +Q0 ~Q0 ·Q0 +Q0 ~Q0 ·Q0 |
40 | 39 | eleq1d 2147 | . . . . . . 7 +Q0 ~Q0 ·Q0 +Q0 ~Q0 ·Q0 |
41 | oveq1 5539 | . . . . . . . . . . . . 13 | |
42 | 41 | oveq1d 5547 | . . . . . . . . . . . 12 |
43 | 42 | opeq1d 3576 | . . . . . . . . . . 11 |
44 | 43 | eceq1d 6165 | . . . . . . . . . 10 |
45 | 44 | oveq1d 5547 | . . . . . . . . 9 |
46 | 45 | oveq2d 5548 | . . . . . . . 8 |
47 | 46 | eleq1d 2147 | . . . . . . 7 |
48 | 40, 47 | anbi12d 456 | . . . . . 6 +Q0 ~Q0 ·Q0 +Q0 ~Q0 ·Q0 |
49 | 48 | rspcev 2701 | . . . . 5 +Q0 ~Q0 ·Q0 +Q0 ~Q0 ·Q0 |
50 | 35, 49 | mpan 414 | . . . 4 +Q0 ~Q0 ·Q0 +Q0 ~Q0 ·Q0 |
51 | 21, 34, 50 | syl6an 1363 | . . 3 +Q0 ~Q0 ·Q0 |
52 | 51 | reximdv 2462 | . 2 +Q0 ~Q0 ·Q0 |
53 | 3, 52 | mpd 13 | 1 +Q0 ~Q0 ·Q0 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 w3a 919 wceq 1284 wcel 1433 wrex 2349 c0 3251 cop 3401 class class class wbr 3785 com 4331 (class class class)co 5532 c1o 6017 c2o 6018 coa 6021 cec 6127 cnpi 6462 clti 6465 ceq 6469 cnq 6470 cplq 6472 cmq 6473 ~Q0 ceq0 6476 Q0cnq0 6477 0Q0c0q0 6478 +Q0 cplq0 6479 ·Q0 cmq0 6480 cnp 6481 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-eprel 4044 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-1o 6024 df-2o 6025 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-mi 6496 df-lti 6497 df-enq 6537 df-nqqs 6538 df-enq0 6614 df-nq0 6615 df-0nq0 6616 df-plq0 6617 df-mq0 6618 df-inp 6656 |
This theorem is referenced by: prarloclem 6691 |
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