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Mirrors > Home > ILE Home > Th. List > 2stropg | Unicode version |
Description: The other slot of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.) |
Ref | Expression |
---|---|
2str.g |
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2str.e |
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2str.l |
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2str.n |
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Ref | Expression |
---|---|
2stropg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2str.e |
. . 3
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2 | 2str.n |
. . 3
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3 | 1, 2 | ndxslid 12023 |
. 2
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4 | 2str.g |
. . 3
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5 | basendxnn 12053 |
. . . . . 6
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6 | 5 | a1i 9 |
. . . . 5
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7 | simpl 108 |
. . . . 5
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8 | opexg 4158 |
. . . . 5
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9 | 6, 7, 8 | syl2anc 409 |
. . . 4
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10 | 1, 2 | ndxarg 12021 |
. . . . . . 7
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11 | 10, 2 | eqeltri 2213 |
. . . . . 6
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12 | 11 | a1i 9 |
. . . . 5
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13 | simpr 109 |
. . . . 5
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14 | opexg 4158 |
. . . . 5
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15 | 12, 13, 14 | syl2anc 409 |
. . . 4
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16 | prexg 4141 |
. . . 4
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17 | 9, 15, 16 | syl2anc 409 |
. . 3
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18 | 4, 17 | eqeltrid 2227 |
. 2
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19 | 5 | nnrei 8753 |
. . . . . 6
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20 | 2str.l |
. . . . . . 7
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21 | basendx 12052 |
. . . . . . 7
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22 | 20, 21, 10 | 3brtr4i 3966 |
. . . . . 6
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23 | 19, 22 | ltneii 7884 |
. . . . 5
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24 | 23 | a1i 9 |
. . . 4
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25 | funprg 5181 |
. . . 4
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26 | 6, 12, 7, 13, 24, 25 | syl221anc 1228 |
. . 3
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27 | 4 | funeqi 5152 |
. . 3
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28 | 26, 27 | sylibr 133 |
. 2
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29 | prid2g 3636 |
. . . 4
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30 | 15, 29 | syl 14 |
. . 3
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31 | 30, 4 | eleqtrrdi 2234 |
. 2
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32 | 3, 18, 28, 31 | strslfvd 12039 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1re 7738 ax-addrcl 7741 ax-pre-ltirr 7756 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-iota 5096 df-fun 5133 df-fv 5139 df-pnf 7826 df-mnf 7827 df-ltxr 7829 df-inn 8745 df-ndx 12001 df-slot 12002 df-base 12004 |
This theorem is referenced by: grpplusgg 12107 eltpsg 12246 |
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