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Mirrors > Home > ILE Home > Th. List > 2strstr1g | GIF version |
Description: A constructed two-slot structure. Version of 2strstrg 12062 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 2-Feb-2023.) |
Ref | Expression |
---|---|
2str1.g | ⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉, 〈𝑁, + 〉} |
2str1.b | ⊢ (Base‘ndx) < 𝑁 |
2str1.n | ⊢ 𝑁 ∈ ℕ |
Ref | Expression |
---|---|
2strstr1g | ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐺 Struct 〈(Base‘ndx), 𝑁〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2str1.g | . . . 4 ⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉, 〈𝑁, + 〉} | |
2 | eqid 2139 | . . . . . . . 8 ⊢ Slot 𝑁 = Slot 𝑁 | |
3 | 2str1.n | . . . . . . . 8 ⊢ 𝑁 ∈ ℕ | |
4 | 2, 3 | ndxarg 11985 | . . . . . . 7 ⊢ (Slot 𝑁‘ndx) = 𝑁 |
5 | 4 | eqcomi 2143 | . . . . . 6 ⊢ 𝑁 = (Slot 𝑁‘ndx) |
6 | 5 | opeq1i 3708 | . . . . 5 ⊢ 〈𝑁, + 〉 = 〈(Slot 𝑁‘ndx), + 〉 |
7 | 6 | preq2i 3604 | . . . 4 ⊢ {〈(Base‘ndx), 𝐵〉, 〈𝑁, + 〉} = {〈(Base‘ndx), 𝐵〉, 〈(Slot 𝑁‘ndx), + 〉} |
8 | 1, 7 | eqtri 2160 | . . 3 ⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉, 〈(Slot 𝑁‘ndx), + 〉} |
9 | basendx 12016 | . . . 4 ⊢ (Base‘ndx) = 1 | |
10 | 2str1.b | . . . 4 ⊢ (Base‘ndx) < 𝑁 | |
11 | 9, 10 | eqbrtrri 3951 | . . 3 ⊢ 1 < 𝑁 |
12 | 8, 2, 11, 3 | 2strstrg 12062 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐺 Struct 〈1, 𝑁〉) |
13 | 9 | opeq1i 3708 | . 2 ⊢ 〈(Base‘ndx), 𝑁〉 = 〈1, 𝑁〉 |
14 | 12, 13 | breqtrrdi 3970 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐺 Struct 〈(Base‘ndx), 𝑁〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 {cpr 3528 〈cop 3530 class class class wbr 3929 ‘cfv 5123 1c1 7624 < clt 7803 ℕcn 8723 Struct cstr 11958 ndxcnx 11959 Slot cslot 11961 Basecbs 11962 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-addcom 7723 ax-addass 7725 ax-distr 7727 ax-i2m1 7728 ax-0lt1 7729 ax-0id 7731 ax-rnegex 7732 ax-cnre 7734 ax-pre-ltirr 7735 ax-pre-ltwlin 7736 ax-pre-lttrn 7737 ax-pre-apti 7738 ax-pre-ltadd 7739 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 df-sub 7938 df-neg 7939 df-inn 8724 df-n0 8981 df-z 9058 df-uz 9330 df-fz 9794 df-struct 11964 df-ndx 11965 df-slot 11966 df-base 11968 |
This theorem is referenced by: 2strbas1g 12066 2strop1g 12067 |
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