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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 12505 |
. 2
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4 | 2str.g |
. . 3
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5 | basendxnn 12536 |
. . . . . 6
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6 | 5 | a1i 9 |
. . . . 5
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7 | simpl 109 |
. . . . 5
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8 | opexg 4243 |
. . . . 5
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9 | 6, 7, 8 | syl2anc 411 |
. . . 4
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10 | 1, 2 | ndxarg 12503 |
. . . . . . 7
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11 | 10, 2 | eqeltri 2262 |
. . . . . 6
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12 | 11 | a1i 9 |
. . . . 5
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13 | simpr 110 |
. . . . 5
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14 | opexg 4243 |
. . . . 5
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15 | 12, 13, 14 | syl2anc 411 |
. . . 4
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16 | prexg 4226 |
. . . 4
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17 | 9, 15, 16 | syl2anc 411 |
. . 3
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18 | 4, 17 | eqeltrid 2276 |
. 2
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19 | 5 | nnrei 8946 |
. . . . . 6
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20 | 2str.l |
. . . . . . 7
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21 | basendx 12535 |
. . . . . . 7
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22 | 20, 21, 10 | 3brtr4i 4048 |
. . . . . 6
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23 | 19, 22 | ltneii 8072 |
. . . . 5
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24 | 23 | a1i 9 |
. . . 4
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25 | funprg 5281 |
. . . 4
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26 | 6, 12, 7, 13, 24, 25 | syl221anc 1260 |
. . 3
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27 | 4 | funeqi 5252 |
. . 3
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28 | 26, 27 | sylibr 134 |
. 2
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29 | prid2g 3712 |
. . . 4
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30 | 15, 29 | syl 14 |
. . 3
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31 | 30, 4 | eleqtrrdi 2283 |
. 2
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32 | 3, 18, 28, 31 | strslfvd 12522 |
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 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-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7920 ax-resscn 7921 ax-1re 7923 ax-addrcl 7926 ax-pre-ltirr 7941 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-iota 5193 df-fun 5233 df-fv 5239 df-pnf 8012 df-mnf 8013 df-ltxr 8015 df-inn 8938 df-ndx 12483 df-slot 12484 df-base 12486 |
This theorem is referenced by: grpplusgg 12605 eltpsg 13937 |
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