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Mirrors > Home > MPE Home > Th. List > 4sqlem4 | Structured version Visualization version Unicode version |
Description: Lemma for 4sq 15668. We can express the four-square property more compactly in terms of gaussian integers, because the norms of gaussian integers are exactly sums of two squares. (Contributed by Mario Carneiro, 14-Jul-2014.) |
Ref | Expression |
---|---|
4sq.1 |
Ref | Expression |
---|---|
4sqlem4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4sq.1 | . . . 4 | |
2 | 1 | 4sqlem2 15653 | . . 3 |
3 | gzreim 15643 | . . . . . . . 8 | |
4 | 3 | adantr 481 | . . . . . . 7 |
5 | gzreim 15643 | . . . . . . . 8 | |
6 | 5 | adantl 482 | . . . . . . 7 |
7 | gzcn 15636 | . . . . . . . . . . . 12 | |
8 | 3, 7 | syl 17 | . . . . . . . . . . 11 |
9 | 8 | absvalsq2d 14182 | . . . . . . . . . 10 |
10 | zre 11381 | . . . . . . . . . . . . 13 | |
11 | zre 11381 | . . . . . . . . . . . . 13 | |
12 | crre 13854 | . . . . . . . . . . . . 13 | |
13 | 10, 11, 12 | syl2an 494 | . . . . . . . . . . . 12 |
14 | 13 | oveq1d 6665 | . . . . . . . . . . 11 |
15 | crim 13855 | . . . . . . . . . . . . 13 | |
16 | 10, 11, 15 | syl2an 494 | . . . . . . . . . . . 12 |
17 | 16 | oveq1d 6665 | . . . . . . . . . . 11 |
18 | 14, 17 | oveq12d 6668 | . . . . . . . . . 10 |
19 | 9, 18 | eqtrd 2656 | . . . . . . . . 9 |
20 | gzcn 15636 | . . . . . . . . . . . 12 | |
21 | 5, 20 | syl 17 | . . . . . . . . . . 11 |
22 | 21 | absvalsq2d 14182 | . . . . . . . . . 10 |
23 | zre 11381 | . . . . . . . . . . . . 13 | |
24 | zre 11381 | . . . . . . . . . . . . 13 | |
25 | crre 13854 | . . . . . . . . . . . . 13 | |
26 | 23, 24, 25 | syl2an 494 | . . . . . . . . . . . 12 |
27 | 26 | oveq1d 6665 | . . . . . . . . . . 11 |
28 | crim 13855 | . . . . . . . . . . . . 13 | |
29 | 23, 24, 28 | syl2an 494 | . . . . . . . . . . . 12 |
30 | 29 | oveq1d 6665 | . . . . . . . . . . 11 |
31 | 27, 30 | oveq12d 6668 | . . . . . . . . . 10 |
32 | 22, 31 | eqtrd 2656 | . . . . . . . . 9 |
33 | 19, 32 | oveqan12d 6669 | . . . . . . . 8 |
34 | 33 | eqcomd 2628 | . . . . . . 7 |
35 | fveq2 6191 | . . . . . . . . . . 11 | |
36 | 35 | oveq1d 6665 | . . . . . . . . . 10 |
37 | 36 | oveq1d 6665 | . . . . . . . . 9 |
38 | 37 | eqeq2d 2632 | . . . . . . . 8 |
39 | fveq2 6191 | . . . . . . . . . . 11 | |
40 | 39 | oveq1d 6665 | . . . . . . . . . 10 |
41 | 40 | oveq2d 6666 | . . . . . . . . 9 |
42 | 41 | eqeq2d 2632 | . . . . . . . 8 |
43 | 38, 42 | rspc2ev 3324 | . . . . . . 7 |
44 | 4, 6, 34, 43 | syl3anc 1326 | . . . . . 6 |
45 | eqeq1 2626 | . . . . . . 7 | |
46 | 45 | 2rexbidv 3057 | . . . . . 6 |
47 | 44, 46 | syl5ibrcom 237 | . . . . 5 |
48 | 47 | rexlimdvva 3038 | . . . 4 |
49 | 48 | rexlimivv 3036 | . . 3 |
50 | 2, 49 | sylbi 207 | . 2 |
51 | 1 | 4sqlem4a 15655 | . . . 4 |
52 | eleq1a 2696 | . . . 4 | |
53 | 51, 52 | syl 17 | . . 3 |
54 | 53 | rexlimivv 3036 | . 2 |
55 | 50, 54 | impbii 199 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 cab 2608 wrex 2913 cfv 5888 (class class class)co 6650 cc 9934 cr 9935 ci 9938 caddc 9939 cmul 9941 c2 11070 cz 11377 cexp 12860 cre 13837 cim 13838 cabs 13974 cgz 15633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-gz 15634 |
This theorem is referenced by: mul4sq 15658 |
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