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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 11766 |
. 2
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4 | 2str.g |
. . 3
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5 | basendxnn 11796 |
. . . . . 6
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6 | 5 | a1i 9 |
. . . . 5
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7 | simpl 108 |
. . . . 5
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8 | opexg 4088 |
. . . . 5
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9 | 6, 7, 8 | syl2anc 406 |
. . . 4
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10 | 1, 2 | ndxarg 11764 |
. . . . . . 7
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11 | 10, 2 | eqeltri 2172 |
. . . . . 6
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12 | 11 | a1i 9 |
. . . . 5
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13 | simpr 109 |
. . . . 5
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14 | opexg 4088 |
. . . . 5
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15 | 12, 13, 14 | syl2anc 406 |
. . . 4
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16 | prexg 4071 |
. . . 4
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17 | 9, 15, 16 | syl2anc 406 |
. . 3
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18 | 4, 17 | syl5eqel 2186 |
. 2
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19 | 5 | nnrei 8587 |
. . . . . 6
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20 | 2str.l |
. . . . . . 7
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21 | basendx 11795 |
. . . . . . 7
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22 | 20, 21, 10 | 3brtr4i 3903 |
. . . . . 6
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23 | 19, 22 | ltneii 7731 |
. . . . 5
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24 | 23 | a1i 9 |
. . . 4
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25 | funprg 5109 |
. . . 4
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26 | 6, 12, 7, 13, 24, 25 | syl221anc 1195 |
. . 3
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27 | 4 | funeqi 5080 |
. . 3
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28 | 26, 27 | sylibr 133 |
. 2
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29 | prid2g 3575 |
. . . 4
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30 | 15, 29 | syl 14 |
. . 3
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31 | 30, 4 | syl6eleqr 2193 |
. 2
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32 | 3, 18, 28, 31 | strslfvd 11782 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-cnex 7586 ax-resscn 7587 ax-1re 7589 ax-addrcl 7592 ax-pre-ltirr 7607 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-rab 2384 df-v 2643 df-sbc 2863 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-br 3876 df-opab 3930 df-mpt 3931 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-iota 5024 df-fun 5061 df-fv 5067 df-pnf 7674 df-mnf 7675 df-ltxr 7677 df-inn 8579 df-ndx 11744 df-slot 11745 df-base 11747 |
This theorem is referenced by: grpplusgg 11850 eltpsg 11989 |
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