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Mirrors > Home > MPE Home > Th. List > ipdiri | Structured version Unicode version |
Description: Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ip1i.1 |
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ip1i.2 |
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ip1i.4 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
ip1i.7 |
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ip1i.9 |
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Ref | Expression |
---|---|
ipdiri |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 6210 |
. . . 4
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2 | 1 | oveq1d 6218 |
. . 3
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3 | oveq1 6210 |
. . . 4
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4 | 3 | oveq1d 6218 |
. . 3
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5 | 2, 4 | eqeq12d 2476 |
. 2
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6 | oveq2 6211 |
. . . 4
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7 | 6 | oveq1d 6218 |
. . 3
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8 | oveq1 6210 |
. . . 4
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9 | 8 | oveq2d 6219 |
. . 3
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10 | 7, 9 | eqeq12d 2476 |
. 2
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11 | oveq2 6211 |
. . 3
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12 | oveq2 6211 |
. . . 4
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13 | oveq2 6211 |
. . . 4
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14 | 12, 13 | oveq12d 6221 |
. . 3
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15 | 11, 14 | eqeq12d 2476 |
. 2
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16 | ip1i.1 |
. . 3
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17 | ip1i.2 |
. . 3
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18 | ip1i.4 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | ip1i.7 |
. . 3
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20 | ip1i.9 |
. . 3
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21 | eqid 2454 |
. . . 4
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22 | 16, 21, 20 | elimph 24399 |
. . 3
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23 | 16, 21, 20 | elimph 24399 |
. . 3
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24 | 16, 21, 20 | elimph 24399 |
. . 3
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25 | 16, 17, 18, 19, 20, 22, 23, 24 | ipdirilem 24408 |
. 2
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26 | 5, 10, 15, 25 | dedth3h 3954 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-rep 4514 ax-sep 4524 ax-nul 4532 ax-pow 4581 ax-pr 4642 ax-un 6485 ax-inf2 7962 ax-cnex 9453 ax-resscn 9454 ax-1cn 9455 ax-icn 9456 ax-addcl 9457 ax-addrcl 9458 ax-mulcl 9459 ax-mulrcl 9460 ax-mulcom 9461 ax-addass 9462 ax-mulass 9463 ax-distr 9464 ax-i2m1 9465 ax-1ne0 9466 ax-1rid 9467 ax-rnegex 9468 ax-rrecex 9469 ax-cnre 9470 ax-pre-lttri 9471 ax-pre-lttrn 9472 ax-pre-ltadd 9473 ax-pre-mulgt0 9474 ax-pre-sup 9475 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-fal 1376 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-nel 2651 df-ral 2804 df-rex 2805 df-reu 2806 df-rmo 2807 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3399 df-dif 3442 df-un 3444 df-in 3446 df-ss 3453 df-pss 3455 df-nul 3749 df-if 3903 df-pw 3973 df-sn 3989 df-pr 3991 df-tp 3993 df-op 3995 df-uni 4203 df-int 4240 df-iun 4284 df-br 4404 df-opab 4462 df-mpt 4463 df-tr 4497 df-eprel 4743 df-id 4747 df-po 4752 df-so 4753 df-fr 4790 df-se 4791 df-we 4792 df-ord 4833 df-on 4834 df-lim 4835 df-suc 4836 df-xp 4957 df-rel 4958 df-cnv 4959 df-co 4960 df-dm 4961 df-rn 4962 df-res 4963 df-ima 4964 df-iota 5492 df-fun 5531 df-fn 5532 df-f 5533 df-f1 5534 df-fo 5535 df-f1o 5536 df-fv 5537 df-isom 5538 df-riota 6164 df-ov 6206 df-oprab 6207 df-mpt2 6208 df-om 6590 df-1st 6690 df-2nd 6691 df-recs 6945 df-rdg 6979 df-1o 7033 df-oadd 7037 df-er 7214 df-en 7424 df-dom 7425 df-sdom 7426 df-fin 7427 df-sup 7806 df-oi 7839 df-card 8224 df-pnf 9535 df-mnf 9536 df-xr 9537 df-ltxr 9538 df-le 9539 df-sub 9712 df-neg 9713 df-div 10109 df-nn 10438 df-2 10495 df-3 10496 df-4 10497 df-n0 10695 df-z 10762 df-uz 10977 df-rp 11107 df-fz 11559 df-fzo 11670 df-seq 11928 df-exp 11987 df-hash 12225 df-cj 12710 df-re 12711 df-im 12712 df-sqr 12846 df-abs 12847 df-clim 13088 df-sum 13286 df-grpo 23857 df-gid 23858 df-ginv 23859 df-ablo 23948 df-vc 24103 df-nv 24149 df-va 24152 df-ba 24153 df-sm 24154 df-0v 24155 df-nmcv 24157 df-dip 24275 df-ph 24392 |
This theorem is referenced by: ipasslem1 24410 ipasslem2 24411 ipasslem11 24419 dipdir 24421 |
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