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Mirrors > Home > ILE Home > Th. List > 2strop1g | GIF version |
Description: The other slot of a constructed two-slot structure. Version of 2stropg 12569 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 12477 | . 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 12570 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐺 Struct 〈(Base‘ndx), 𝑁〉) |
7 | simpr 110 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → + ∈ 𝑊) | |
8 | opexg 4226 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ + ∈ 𝑊) → 〈𝑁, + 〉 ∈ V) | |
9 | 2, 7, 8 | sylancr 414 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 〈𝑁, + 〉 ∈ V) |
10 | prid2g 3697 | . . . 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 12475 | . . . 4 ⊢ (𝐸‘ndx) = 𝑁 |
13 | 12 | opeq1i 3780 | . . 3 ⊢ 〈(𝐸‘ndx), + 〉 = 〈𝑁, + 〉 |
14 | 11, 13, 4 | 3eltr4g 2263 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 〈(𝐸‘ndx), + 〉 ∈ 𝐺) |
15 | 3, 6, 7, 14 | opelstrsl 12563 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → + = (𝐸‘𝐺)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 Vcvv 2737 {cpr 3593 〈cop 3595 class class class wbr 4001 ‘cfv 5213 < clt 7986 ℕcn 8913 ndxcnx 12449 Slot cslot 12451 Basecbs 12452 |
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 4119 ax-pow 4172 ax-pr 4207 ax-un 4431 ax-setind 4534 ax-cnex 7897 ax-resscn 7898 ax-1cn 7899 ax-1re 7900 ax-icn 7901 ax-addcl 7902 ax-addrcl 7903 ax-mulcl 7904 ax-addcom 7906 ax-addass 7908 ax-distr 7910 ax-i2m1 7911 ax-0lt1 7912 ax-0id 7914 ax-rnegex 7915 ax-cnre 7917 ax-pre-ltirr 7918 ax-pre-ltwlin 7919 ax-pre-lttrn 7920 ax-pre-apti 7921 ax-pre-ltadd 7922 |
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 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3809 df-int 3844 df-br 4002 df-opab 4063 df-mpt 4064 df-id 4291 df-xp 4630 df-rel 4631 df-cnv 4632 df-co 4633 df-dm 4634 df-rn 4635 df-res 4636 df-ima 4637 df-iota 5175 df-fun 5215 df-fn 5216 df-f 5217 df-fv 5221 df-riota 5826 df-ov 5873 df-oprab 5874 df-mpo 5875 df-pnf 7988 df-mnf 7989 df-xr 7990 df-ltxr 7991 df-le 7992 df-sub 8124 df-neg 8125 df-inn 8914 df-n0 9171 df-z 9248 df-uz 9523 df-fz 10003 df-struct 12454 df-ndx 12455 df-slot 12456 df-base 12458 |
This theorem is referenced by: (None) |
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