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Mirrors > Home > ILE Home > Th. List > 2strstrg | GIF version |
Description: A constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.) |
Ref | Expression |
---|---|
2str.g | β’ πΊ = {β¨(Baseβndx), π΅β©, β¨(πΈβndx), + β©} |
2str.e | β’ πΈ = Slot π |
2str.l | β’ 1 < π |
2str.n | β’ π β β |
Ref | Expression |
---|---|
2strstrg | β’ ((π΅ β π β§ + β π) β πΊ Struct β¨1, πβ©) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2str.g | . 2 β’ πΊ = {β¨(Baseβndx), π΅β©, β¨(πΈβndx), + β©} | |
2 | 1nn 8933 | . . 3 β’ 1 β β | |
3 | basendx 12520 | . . 3 β’ (Baseβndx) = 1 | |
4 | 2str.l | . . 3 β’ 1 < π | |
5 | 2str.n | . . 3 β’ π β β | |
6 | 2str.e | . . . 4 β’ πΈ = Slot π | |
7 | 6, 5 | ndxarg 12488 | . . 3 β’ (πΈβndx) = π |
8 | 2, 3, 4, 5, 7 | strle2g 12569 | . 2 β’ ((π΅ β π β§ + β π) β {β¨(Baseβndx), π΅β©, β¨(πΈβndx), + β©} Struct β¨1, πβ©) |
9 | 1, 8 | eqbrtrid 4040 | 1 β’ ((π΅ β π β§ + β π) β πΊ Struct β¨1, πβ©) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 = wceq 1353 β wcel 2148 {cpr 3595 β¨cop 3597 class class class wbr 4005 βcfv 5218 1c1 7815 < clt 7995 βcn 8922 Struct cstr 12461 ndxcnx 12462 Slot cslot 12464 Basecbs 12465 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7905 ax-resscn 7906 ax-1cn 7907 ax-1re 7908 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-addcom 7914 ax-addass 7916 ax-distr 7918 ax-i2m1 7919 ax-0lt1 7920 ax-0id 7922 ax-rnegex 7923 ax-cnre 7925 ax-pre-ltirr 7926 ax-pre-ltwlin 7927 ax-pre-lttrn 7928 ax-pre-apti 7929 ax-pre-ltadd 7930 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-fv 5226 df-riota 5834 df-ov 5881 df-oprab 5882 df-mpo 5883 df-pnf 7997 df-mnf 7998 df-xr 7999 df-ltxr 8000 df-le 8001 df-sub 8133 df-neg 8134 df-inn 8923 df-n0 9180 df-z 9257 df-uz 9532 df-fz 10012 df-struct 12467 df-ndx 12468 df-slot 12469 df-base 12471 |
This theorem is referenced by: 2strstr1g 12583 grpstrg 12587 |
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