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| Mirrors > Home > ILE Home > Th. List > recidpirqlemcalc | Unicode version | ||
| Description: Lemma for recidpirq 7971. 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 7667 |
. . . . . 6
| |
| 3 | 2 | a1i 9 |
. . . . 5
|
| 4 | addclpr 7650 |
. . . . 5
| |
| 5 | 1, 3, 4 | syl2anc 411 |
. . . 4
|
| 6 | recidpirqlemcalc.b |
. . . . 5
| |
| 7 | addclpr 7650 |
. . . . 5
| |
| 8 | 6, 3, 7 | syl2anc 411 |
. . . 4
|
| 9 | addclpr 7650 |
. . . 4
| |
| 10 | 5, 8, 9 | syl2anc 411 |
. . 3
|
| 11 | addassprg 7692 |
. . 3
| |
| 12 | 10, 3, 3, 11 | syl3anc 1250 |
. 2
|
| 13 | distrprg 7701 |
. . . . . . 7
| |
| 14 | 5, 6, 3, 13 | syl3anc 1250 |
. . . . . 6
|
| 15 | 1idpr 7705 |
. . . . . . . 8
| |
| 16 | 5, 15 | syl 14 |
. . . . . . 7
|
| 17 | 16 | oveq2d 5960 |
. . . . . 6
|
| 18 | mulcomprg 7693 |
. . . . . . . . 9
| |
| 19 | 5, 6, 18 | syl2anc 411 |
. . . . . . . 8
|
| 20 | distrprg 7701 |
. . . . . . . . 9
| |
| 21 | 6, 1, 3, 20 | syl3anc 1250 |
. . . . . . . 8
|
| 22 | mulcomprg 7693 |
. . . . . . . . . . 11
| |
| 23 | 6, 1, 22 | syl2anc 411 |
. . . . . . . . . 10
|
| 24 | recidpirqlemcalc.rec |
. . . . . . . . . 10
| |
| 25 | 23, 24 | eqtrd 2238 |
. . . . . . . . 9
|
| 26 | 1idpr 7705 |
. . . . . . . . . 10
| |
| 27 | 6, 26 | syl 14 |
. . . . . . . . 9
|
| 28 | 25, 27 | oveq12d 5962 |
. . . . . . . 8
|
| 29 | 19, 21, 28 | 3eqtrd 2242 |
. . . . . . 7
|
| 30 | 29 | oveq1d 5959 |
. . . . . 6
|
| 31 | 14, 17, 30 | 3eqtrd 2242 |
. . . . 5
|
| 32 | 1idpr 7705 |
. . . . . 6
| |
| 33 | 2, 32 | mp1i 10 |
. . . . 5
|
| 34 | 31, 33 | oveq12d 5962 |
. . . 4
|
| 35 | addcomprg 7691 |
. . . . . . . 8
| |
| 36 | 3, 6, 35 | syl2anc 411 |
. . . . . . 7
|
| 37 | 36 | oveq1d 5959 |
. . . . . 6
|
| 38 | addcomprg 7691 |
. . . . . . 7
| |
| 39 | 8, 5, 38 | syl2anc 411 |
. . . . . 6
|
| 40 | 37, 39 | eqtrd 2238 |
. . . . 5
|
| 41 | 40 | oveq1d 5959 |
. . . 4
|
| 42 | 34, 41 | eqtrd 2238 |
. . 3
|
| 43 | 42 | oveq1d 5959 |
. 2
|
| 44 | mulcomprg 7693 |
. . . . . 6
| |
| 45 | 3, 8, 44 | syl2anc 411 |
. . . . 5
|
| 46 | 1idpr 7705 |
. . . . . 6
| |
| 47 | 8, 46 | syl 14 |
. . . . 5
|
| 48 | 45, 47 | eqtrd 2238 |
. . . 4
|
| 49 | 16, 48 | oveq12d 5962 |
. . 3
|
| 50 | 49 | oveq1d 5959 |
. 2
|
| 51 | 12, 43, 50 | 3eqtr4d 2248 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-coll 4159 ax-sep 4162 ax-nul 4170 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-iinf 4636 |
| This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-ral 2489 df-rex 2490 df-reu 2491 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-iun 3929 df-br 4045 df-opab 4106 df-mpt 4107 df-tr 4143 df-eprel 4336 df-id 4340 df-po 4343 df-iso 4344 df-iord 4413 df-on 4415 df-suc 4418 df-iom 4639 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-f1 5276 df-fo 5277 df-f1o 5278 df-fv 5279 df-ov 5947 df-oprab 5948 df-mpo 5949 df-1st 6226 df-2nd 6227 df-recs 6391 df-irdg 6456 df-1o 6502 df-2o 6503 df-oadd 6506 df-omul 6507 df-er 6620 df-ec 6622 df-qs 6626 df-ni 7417 df-pli 7418 df-mi 7419 df-lti 7420 df-plpq 7457 df-mpq 7458 df-enq 7460 df-nqqs 7461 df-plqqs 7462 df-mqqs 7463 df-1nqqs 7464 df-rq 7465 df-ltnqqs 7466 df-enq0 7537 df-nq0 7538 df-0nq0 7539 df-plq0 7540 df-mq0 7541 df-inp 7579 df-i1p 7580 df-iplp 7581 df-imp 7582 |
| This theorem is referenced by: recidpirq 7971 |
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