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Mirrors > Home > ILE Home > Th. List > 2strstr1g | GIF version |
Description: A constructed two-slot structure. Version of 2strstrg 12518 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 2170 | . . . . . . . 8 ⊢ Slot 𝑁 = Slot 𝑁 | |
3 | 2str1.n | . . . . . . . 8 ⊢ 𝑁 ∈ ℕ | |
4 | 2, 3 | ndxarg 12439 | . . . . . . 7 ⊢ (Slot 𝑁‘ndx) = 𝑁 |
5 | 4 | eqcomi 2174 | . . . . . 6 ⊢ 𝑁 = (Slot 𝑁‘ndx) |
6 | 5 | opeq1i 3768 | . . . . 5 ⊢ 〈𝑁, + 〉 = 〈(Slot 𝑁‘ndx), + 〉 |
7 | 6 | preq2i 3664 | . . . 4 ⊢ {〈(Base‘ndx), 𝐵〉, 〈𝑁, + 〉} = {〈(Base‘ndx), 𝐵〉, 〈(Slot 𝑁‘ndx), + 〉} |
8 | 1, 7 | eqtri 2191 | . . 3 ⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉, 〈(Slot 𝑁‘ndx), + 〉} |
9 | basendx 12470 | . . . 4 ⊢ (Base‘ndx) = 1 | |
10 | 2str1.b | . . . 4 ⊢ (Base‘ndx) < 𝑁 | |
11 | 9, 10 | eqbrtrri 4012 | . . 3 ⊢ 1 < 𝑁 |
12 | 8, 2, 11, 3 | 2strstrg 12518 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐺 Struct 〈1, 𝑁〉) |
13 | 9 | opeq1i 3768 | . 2 ⊢ 〈(Base‘ndx), 𝑁〉 = 〈1, 𝑁〉 |
14 | 12, 13 | breqtrrdi 4031 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐺 Struct 〈(Base‘ndx), 𝑁〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∈ wcel 2141 {cpr 3584 〈cop 3586 class class class wbr 3989 ‘cfv 5198 1c1 7775 < clt 7954 ℕcn 8878 Struct cstr 12412 ndxcnx 12413 Slot cslot 12415 Basecbs 12416 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-n0 9136 df-z 9213 df-uz 9488 df-fz 9966 df-struct 12418 df-ndx 12419 df-slot 12420 df-base 12422 |
This theorem is referenced by: 2strbas1g 12522 2strop1g 12523 |
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