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Mirrors > Home > ILE Home > Th. List > recidpirqlemcalc | Unicode version |
Description: Lemma for recidpirq 7790. Rearranging some of the expressions. (Contributed by Jim Kingdon, 17-Jul-2021.) |
Ref | Expression |
---|---|
recidpirqlemcalc.a | |
recidpirqlemcalc.b | |
recidpirqlemcalc.rec |
Ref | Expression |
---|---|
recidpirqlemcalc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recidpirqlemcalc.a | . . . . 5 | |
2 | 1pr 7486 | . . . . . 6 | |
3 | 2 | a1i 9 | . . . . 5 |
4 | addclpr 7469 | . . . . 5 | |
5 | 1, 3, 4 | syl2anc 409 | . . . 4 |
6 | recidpirqlemcalc.b | . . . . 5 | |
7 | addclpr 7469 | . . . . 5 | |
8 | 6, 3, 7 | syl2anc 409 | . . . 4 |
9 | addclpr 7469 | . . . 4 | |
10 | 5, 8, 9 | syl2anc 409 | . . 3 |
11 | addassprg 7511 | . . 3 | |
12 | 10, 3, 3, 11 | syl3anc 1227 | . 2 |
13 | distrprg 7520 | . . . . . . 7 | |
14 | 5, 6, 3, 13 | syl3anc 1227 | . . . . . 6 |
15 | 1idpr 7524 | . . . . . . . 8 | |
16 | 5, 15 | syl 14 | . . . . . . 7 |
17 | 16 | oveq2d 5852 | . . . . . 6 |
18 | mulcomprg 7512 | . . . . . . . . 9 | |
19 | 5, 6, 18 | syl2anc 409 | . . . . . . . 8 |
20 | distrprg 7520 | . . . . . . . . 9 | |
21 | 6, 1, 3, 20 | syl3anc 1227 | . . . . . . . 8 |
22 | mulcomprg 7512 | . . . . . . . . . . 11 | |
23 | 6, 1, 22 | syl2anc 409 | . . . . . . . . . 10 |
24 | recidpirqlemcalc.rec | . . . . . . . . . 10 | |
25 | 23, 24 | eqtrd 2197 | . . . . . . . . 9 |
26 | 1idpr 7524 | . . . . . . . . . 10 | |
27 | 6, 26 | syl 14 | . . . . . . . . 9 |
28 | 25, 27 | oveq12d 5854 | . . . . . . . 8 |
29 | 19, 21, 28 | 3eqtrd 2201 | . . . . . . 7 |
30 | 29 | oveq1d 5851 | . . . . . 6 |
31 | 14, 17, 30 | 3eqtrd 2201 | . . . . 5 |
32 | 1idpr 7524 | . . . . . 6 | |
33 | 2, 32 | mp1i 10 | . . . . 5 |
34 | 31, 33 | oveq12d 5854 | . . . 4 |
35 | addcomprg 7510 | . . . . . . . 8 | |
36 | 3, 6, 35 | syl2anc 409 | . . . . . . 7 |
37 | 36 | oveq1d 5851 | . . . . . 6 |
38 | addcomprg 7510 | . . . . . . 7 | |
39 | 8, 5, 38 | syl2anc 409 | . . . . . 6 |
40 | 37, 39 | eqtrd 2197 | . . . . 5 |
41 | 40 | oveq1d 5851 | . . . 4 |
42 | 34, 41 | eqtrd 2197 | . . 3 |
43 | 42 | oveq1d 5851 | . 2 |
44 | mulcomprg 7512 | . . . . . 6 | |
45 | 3, 8, 44 | syl2anc 409 | . . . . 5 |
46 | 1idpr 7524 | . . . . . 6 | |
47 | 8, 46 | syl 14 | . . . . 5 |
48 | 45, 47 | eqtrd 2197 | . . . 4 |
49 | 16, 48 | oveq12d 5854 | . . 3 |
50 | 49 | oveq1d 5851 | . 2 |
51 | 12, 43, 50 | 3eqtr4d 2207 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1342 wcel 2135 (class class class)co 5836 cnp 7223 c1p 7224 cpp 7225 cmp 7226 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-eprel 4261 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-1o 6375 df-2o 6376 df-oadd 6379 df-omul 6380 df-er 6492 df-ec 6494 df-qs 6498 df-ni 7236 df-pli 7237 df-mi 7238 df-lti 7239 df-plpq 7276 df-mpq 7277 df-enq 7279 df-nqqs 7280 df-plqqs 7281 df-mqqs 7282 df-1nqqs 7283 df-rq 7284 df-ltnqqs 7285 df-enq0 7356 df-nq0 7357 df-0nq0 7358 df-plq0 7359 df-mq0 7360 df-inp 7398 df-i1p 7399 df-iplp 7400 df-imp 7401 |
This theorem is referenced by: recidpirq 7790 |
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