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Mirrors > Home > ILE Home > Th. List > sqabsadd | Unicode version |
Description: Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason] p. 133. (Contributed by NM, 21-Jan-2007.) |
Ref | Expression |
---|---|
sqabsadd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cjadd 10656 | . . . 4 | |
2 | 1 | oveq2d 5790 | . . 3 |
3 | cjcl 10620 | . . . . 5 | |
4 | cjcl 10620 | . . . . 5 | |
5 | 3, 4 | anim12i 336 | . . . 4 |
6 | muladd 8146 | . . . 4 | |
7 | 5, 6 | mpdan 417 | . . 3 |
8 | 2, 7 | eqtrd 2172 | . 2 |
9 | addcl 7745 | . . 3 | |
10 | absvalsq 10825 | . . 3 | |
11 | 9, 10 | syl 14 | . 2 |
12 | absvalsq 10825 | . . . 4 | |
13 | absvalsq 10825 | . . . . 5 | |
14 | mulcom 7749 | . . . . . 6 | |
15 | 4, 14 | mpdan 417 | . . . . 5 |
16 | 13, 15 | eqtrd 2172 | . . . 4 |
17 | 12, 16 | oveqan12d 5793 | . . 3 |
18 | mulcl 7747 | . . . . . 6 | |
19 | 4, 18 | sylan2 284 | . . . . 5 |
20 | 19 | addcjd 10729 | . . . 4 |
21 | cjmul 10657 | . . . . . . 7 | |
22 | 4, 21 | sylan2 284 | . . . . . 6 |
23 | cjcj 10655 | . . . . . . . 8 | |
24 | 23 | adantl 275 | . . . . . . 7 |
25 | 24 | oveq2d 5790 | . . . . . 6 |
26 | 22, 25 | eqtrd 2172 | . . . . 5 |
27 | 26 | oveq2d 5790 | . . . 4 |
28 | 20, 27 | eqtr3d 2174 | . . 3 |
29 | 17, 28 | oveq12d 5792 | . 2 |
30 | 8, 11, 29 | 3eqtr4d 2182 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cfv 5123 (class class class)co 5774 cc 7618 caddc 7623 cmul 7625 c2 8771 cexp 10292 ccj 10611 cre 10612 cabs 10769 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-rp 9442 df-seqfrec 10219 df-exp 10293 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 |
This theorem is referenced by: abstri 10876 sqabsaddi 10924 |
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