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Mirrors > Home > ILE Home > Th. List > 2idlval | Unicode version |
Description: Definition of a two-sided ideal. (Contributed by Mario Carneiro, 14-Jun-2015.) |
Ref | Expression |
---|---|
2idlval.i |
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2idlval.o |
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2idlval.j |
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2idlval.t |
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Ref | Expression |
---|---|
2idlval |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2idlval.t |
. . . 4
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2 | 1 | 2idlmex 14033 |
. . 3
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3 | elinel1 3349 |
. . . 4
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4 | 2idlval.i |
. . . . 5
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5 | 4 | lidlmex 14007 |
. . . 4
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6 | 3, 5 | syl 14 |
. . 3
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7 | lidlex 14005 |
. . . . . . . 8
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8 | 4, 7 | eqeltrid 2283 |
. . . . . . 7
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9 | inex1g 4169 |
. . . . . . 7
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10 | 8, 9 | syl 14 |
. . . . . 6
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11 | fveq2 5558 |
. . . . . . . . 9
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12 | 11, 4 | eqtr4di 2247 |
. . . . . . . 8
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13 | fveq2 5558 |
. . . . . . . . . . 11
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14 | 2idlval.o |
. . . . . . . . . . 11
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15 | 13, 14 | eqtr4di 2247 |
. . . . . . . . . 10
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16 | 15 | fveq2d 5562 |
. . . . . . . . 9
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17 | 2idlval.j |
. . . . . . . . 9
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18 | 16, 17 | eqtr4di 2247 |
. . . . . . . 8
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19 | 12, 18 | ineq12d 3365 |
. . . . . . 7
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20 | df-2idl 14032 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
21 | 19, 20 | fvmptg 5637 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 10, 21 | mpdan 421 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 1, 22 | eqtrid 2241 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 23 | eleq2d 2266 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 2, 6, 24 | pm5.21nii 705 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 25 | eqriv 2193 |
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-in1 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-cnex 7968 ax-resscn 7969 ax-1re 7971 ax-addrcl 7974 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-id 4328 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-ov 5925 df-oprab 5926 df-mpo 5927 df-inn 8988 df-2 9046 df-3 9047 df-4 9048 df-5 9049 df-6 9050 df-7 9051 df-8 9052 df-ndx 12657 df-slot 12658 df-base 12660 df-sets 12661 df-iress 12662 df-mulr 12745 df-sca 12747 df-vsca 12748 df-ip 12749 df-lssm 13885 df-sra 13967 df-rgmod 13968 df-lidl 14001 df-2idl 14032 |
This theorem is referenced by: (None) |
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