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Mirrors > Home > ILE Home > Th. List > mul4sq | Unicode version |
Description: Euler's four-square identity: The product of two sums of four squares is also a sum of four squares. This is usually quoted as an explicit formula involving eight real variables; we save some time by working with complex numbers (gaussian integers) instead, so that we only have to work with four variables, and also hiding the actual formula for the product in the proof of mul4sqlem 12334. (For the curious, the explicit formula that is used is .) (Contributed by Mario Carneiro, 14-Jul-2014.) |
Ref | Expression |
---|---|
4sq.1 |
Ref | Expression |
---|---|
mul4sq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4sq.1 | . . 3 | |
2 | 1 | 4sqlem4 12333 | . 2 |
3 | 1 | 4sqlem4 12333 | . 2 |
4 | reeanv 2639 | . . 3 | |
5 | reeanv 2639 | . . . . 5 | |
6 | simpll 524 | . . . . . . . . . . . . 13 | |
7 | gzabssqcl 12322 | . . . . . . . . . . . . 13 | |
8 | 6, 7 | syl 14 | . . . . . . . . . . . 12 |
9 | simprl 526 | . . . . . . . . . . . . 13 | |
10 | gzabssqcl 12322 | . . . . . . . . . . . . 13 | |
11 | 9, 10 | syl 14 | . . . . . . . . . . . 12 |
12 | 8, 11 | nn0addcld 9181 | . . . . . . . . . . 11 |
13 | 12 | nn0cnd 9179 | . . . . . . . . . 10 |
14 | 13 | div1d 8686 | . . . . . . . . 9 |
15 | simplr 525 | . . . . . . . . . . . . 13 | |
16 | gzabssqcl 12322 | . . . . . . . . . . . . 13 | |
17 | 15, 16 | syl 14 | . . . . . . . . . . . 12 |
18 | simprr 527 | . . . . . . . . . . . . 13 | |
19 | gzabssqcl 12322 | . . . . . . . . . . . . 13 | |
20 | 18, 19 | syl 14 | . . . . . . . . . . . 12 |
21 | 17, 20 | nn0addcld 9181 | . . . . . . . . . . 11 |
22 | 21 | nn0cnd 9179 | . . . . . . . . . 10 |
23 | 22 | div1d 8686 | . . . . . . . . 9 |
24 | 14, 23 | oveq12d 5869 | . . . . . . . 8 |
25 | eqid 2170 | . . . . . . . . 9 | |
26 | eqid 2170 | . . . . . . . . 9 | |
27 | 1nn 8878 | . . . . . . . . . 10 | |
28 | 27 | a1i 9 | . . . . . . . . 9 |
29 | gzsubcl 12321 | . . . . . . . . . . . . 13 | |
30 | 29 | adantr 274 | . . . . . . . . . . . 12 |
31 | gzcn 12313 | . . . . . . . . . . . 12 | |
32 | 30, 31 | syl 14 | . . . . . . . . . . 11 |
33 | 32 | div1d 8686 | . . . . . . . . . 10 |
34 | 33, 30 | eqeltrd 2247 | . . . . . . . . 9 |
35 | gzsubcl 12321 | . . . . . . . . . . . . 13 | |
36 | 35 | adantl 275 | . . . . . . . . . . . 12 |
37 | gzcn 12313 | . . . . . . . . . . . 12 | |
38 | 36, 37 | syl 14 | . . . . . . . . . . 11 |
39 | 38 | div1d 8686 | . . . . . . . . . 10 |
40 | 39, 36 | eqeltrd 2247 | . . . . . . . . 9 |
41 | 14, 12 | eqeltrd 2247 | . . . . . . . . 9 |
42 | 1, 6, 9, 15, 18, 25, 26, 28, 34, 40, 41 | mul4sqlem 12334 | . . . . . . . 8 |
43 | 24, 42 | eqeltrrd 2248 | . . . . . . 7 |
44 | oveq12 5860 | . . . . . . . 8 | |
45 | 44 | eleq1d 2239 | . . . . . . 7 |
46 | 43, 45 | syl5ibrcom 156 | . . . . . 6 |
47 | 46 | rexlimdvva 2595 | . . . . 5 |
48 | 5, 47 | syl5bir 152 | . . . 4 |
49 | 48 | rexlimivv 2593 | . . 3 |
50 | 4, 49 | sylbir 134 | . 2 |
51 | 2, 3, 50 | syl2anb 289 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wcel 2141 cab 2156 wrex 2449 cfv 5196 (class class class)co 5851 cc 7761 c1 7764 caddc 7766 cmul 7768 cmin 8079 cdiv 8578 cn 8867 c2 8918 cn0 9124 cz 9201 cexp 10464 cabs 10950 cgz 12310 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7854 ax-resscn 7855 ax-1cn 7856 ax-1re 7857 ax-icn 7858 ax-addcl 7859 ax-addrcl 7860 ax-mulcl 7861 ax-mulrcl 7862 ax-addcom 7863 ax-mulcom 7864 ax-addass 7865 ax-mulass 7866 ax-distr 7867 ax-i2m1 7868 ax-0lt1 7869 ax-1rid 7870 ax-0id 7871 ax-rnegex 7872 ax-precex 7873 ax-cnre 7874 ax-pre-ltirr 7875 ax-pre-ltwlin 7876 ax-pre-lttrn 7877 ax-pre-apti 7878 ax-pre-ltadd 7879 ax-pre-mulgt0 7880 ax-pre-mulext 7881 ax-arch 7882 ax-caucvg 7883 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-recs 6282 df-frec 6368 df-pnf 7945 df-mnf 7946 df-xr 7947 df-ltxr 7948 df-le 7949 df-sub 8081 df-neg 8082 df-reap 8483 df-ap 8490 df-div 8579 df-inn 8868 df-2 8926 df-3 8927 df-4 8928 df-n0 9125 df-z 9202 df-uz 9477 df-rp 9600 df-seqfrec 10391 df-exp 10465 df-cj 10795 df-re 10796 df-im 10797 df-rsqrt 10951 df-abs 10952 df-gz 12311 |
This theorem is referenced by: (None) |
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