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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 | |
2str.e | Slot |
2str.l | |
2str.n |
Ref | Expression |
---|---|
2stropg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2str.e | . . 3 Slot | |
2 | 2str.n | . . 3 | |
3 | 1, 2 | ndxslid 12382 | . 2 Slot |
4 | 2str.g | . . 3 | |
5 | basendxnn 12412 | . . . . . 6 | |
6 | 5 | a1i 9 | . . . . 5 |
7 | simpl 108 | . . . . 5 | |
8 | opexg 4201 | . . . . 5 | |
9 | 6, 7, 8 | syl2anc 409 | . . . 4 |
10 | 1, 2 | ndxarg 12380 | . . . . . . 7 |
11 | 10, 2 | eqeltri 2237 | . . . . . 6 |
12 | 11 | a1i 9 | . . . . 5 |
13 | simpr 109 | . . . . 5 | |
14 | opexg 4201 | . . . . 5 | |
15 | 12, 13, 14 | syl2anc 409 | . . . 4 |
16 | prexg 4184 | . . . 4 | |
17 | 9, 15, 16 | syl2anc 409 | . . 3 |
18 | 4, 17 | eqeltrid 2251 | . 2 |
19 | 5 | nnrei 8858 | . . . . . 6 |
20 | 2str.l | . . . . . . 7 | |
21 | basendx 12411 | . . . . . . 7 | |
22 | 20, 21, 10 | 3brtr4i 4007 | . . . . . 6 |
23 | 19, 22 | ltneii 7987 | . . . . 5 |
24 | 23 | a1i 9 | . . . 4 |
25 | funprg 5233 | . . . 4 | |
26 | 6, 12, 7, 13, 24, 25 | syl221anc 1238 | . . 3 |
27 | 4 | funeqi 5204 | . . 3 |
28 | 26, 27 | sylibr 133 | . 2 |
29 | prid2g 3676 | . . . 4 | |
30 | 15, 29 | syl 14 | . . 3 |
31 | 30, 4 | eleqtrrdi 2258 | . 2 |
32 | 3, 18, 28, 31 | strslfvd 12398 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wcel 2135 wne 2334 cvv 2722 cpr 3572 cop 3574 class class class wbr 3977 wfun 5177 cfv 5183 c1 7746 clt 7925 cn 8849 cnx 12354 Slot cslot 12356 cbs 12357 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 ax-cnex 7836 ax-resscn 7837 ax-1re 7839 ax-addrcl 7842 ax-pre-ltirr 7857 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2724 df-sbc 2948 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-nul 3406 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-int 3820 df-br 3978 df-opab 4039 df-mpt 4040 df-id 4266 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-rn 4610 df-res 4611 df-iota 5148 df-fun 5185 df-fv 5191 df-pnf 7927 df-mnf 7928 df-ltxr 7930 df-inn 8850 df-ndx 12360 df-slot 12361 df-base 12363 |
This theorem is referenced by: grpplusgg 12466 eltpsg 12605 |
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