![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > 2strop1g | GIF version |
Description: The other slot of a constructed two-slot structure. Version of 2stropg 12520 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 |
2str1.e | ⊢ 𝐸 = Slot 𝑁 |
Ref | Expression |
---|---|
2strop1g | ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → + = (𝐸‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2str1.e | . . 3 ⊢ 𝐸 = Slot 𝑁 | |
2 | 2str1.n | . . 3 ⊢ 𝑁 ∈ ℕ | |
3 | 1, 2 | ndxslid 12441 | . 2 ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) |
4 | 2str1.g | . . 3 ⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉, 〈𝑁, + 〉} | |
5 | 2str1.b | . . 3 ⊢ (Base‘ndx) < 𝑁 | |
6 | 4, 5, 2 | 2strstr1g 12521 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐺 Struct 〈(Base‘ndx), 𝑁〉) |
7 | simpr 109 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → + ∈ 𝑊) | |
8 | opexg 4213 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ + ∈ 𝑊) → 〈𝑁, + 〉 ∈ V) | |
9 | 2, 7, 8 | sylancr 412 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 〈𝑁, + 〉 ∈ V) |
10 | prid2g 3688 | . . . 4 ⊢ (〈𝑁, + 〉 ∈ V → 〈𝑁, + 〉 ∈ {〈(Base‘ndx), 𝐵〉, 〈𝑁, + 〉}) | |
11 | 9, 10 | syl 14 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 〈𝑁, + 〉 ∈ {〈(Base‘ndx), 𝐵〉, 〈𝑁, + 〉}) |
12 | 1, 2 | ndxarg 12439 | . . . 4 ⊢ (𝐸‘ndx) = 𝑁 |
13 | 12 | opeq1i 3768 | . . 3 ⊢ 〈(𝐸‘ndx), + 〉 = 〈𝑁, + 〉 |
14 | 11, 13, 4 | 3eltr4g 2256 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 〈(𝐸‘ndx), + 〉 ∈ 𝐺) |
15 | 3, 6, 7, 14 | opelstrsl 12514 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → + = (𝐸‘𝐺)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∈ wcel 2141 Vcvv 2730 {cpr 3584 〈cop 3586 class class class wbr 3989 ‘cfv 5198 < clt 7954 ℕcn 8878 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: (None) |
Copyright terms: Public domain | W3C validator |