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Mirrors > Home > ILE Home > Th. List > mulrid | Unicode version |
Description: ![]() |
Ref | Expression |
---|---|
mulrid |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnre 7982 |
. 2
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2 | recn 7973 |
. . . . . 6
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3 | ax-icn 7935 |
. . . . . . 7
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4 | recn 7973 |
. . . . . . 7
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5 | mulcl 7967 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | 3, 4, 5 | sylancr 414 |
. . . . . 6
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7 | ax-1cn 7933 |
. . . . . . 7
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8 | adddir 7977 |
. . . . . . 7
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9 | 7, 8 | mp3an3 1337 |
. . . . . 6
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10 | 2, 6, 9 | syl2an 289 |
. . . . 5
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11 | ax-1rid 7947 |
. . . . . 6
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12 | mulass 7971 |
. . . . . . . . 9
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13 | 3, 7, 12 | mp3an13 1339 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 4, 13 | syl 14 |
. . . . . . 7
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15 | ax-1rid 7947 |
. . . . . . . 8
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16 | 15 | oveq2d 5911 |
. . . . . . 7
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17 | 14, 16 | eqtrd 2222 |
. . . . . 6
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18 | 11, 17 | oveqan12d 5914 |
. . . . 5
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19 | 10, 18 | eqtrd 2222 |
. . . 4
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20 | oveq1 5902 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
21 | id 19 |
. . . . 5
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22 | 20, 21 | eqeq12d 2204 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 19, 22 | syl5ibrcom 157 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 23 | rexlimivv 2613 |
. 2
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25 | 1, 24 | syl 14 |
1
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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-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-resscn 7932 ax-1cn 7933 ax-icn 7935 ax-addcl 7936 ax-mulcl 7938 ax-mulcom 7941 ax-mulass 7943 ax-distr 7944 ax-1rid 7947 ax-cnre 7951 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-iota 5196 df-fv 5243 df-ov 5898 |
This theorem is referenced by: mullid 7984 mulid1i 7988 mulridd 8003 muleqadd 8654 divdivap1 8709 conjmulap 8715 nnmulcl 8969 expmul 10595 binom21 10663 binom2sub1 10665 bernneq 10671 hashiun 11517 fproddccvg 11611 prodmodclem2a 11615 efexp 11721 cncrng 13869 cnfld1 13872 ecxp 14774 lgsdilem2 14890 |
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