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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 11745 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 11668 | . 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 11746 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐺 Struct 〈(Base‘ndx), 𝑁〉) |
7 | simpr 109 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → + ∈ 𝑊) | |
8 | opexg 4079 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ + ∈ 𝑊) → 〈𝑁, + 〉 ∈ V) | |
9 | 2, 7, 8 | sylancr 406 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 〈𝑁, + 〉 ∈ V) |
10 | prid2g 3567 | . . . 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 11666 | . . . 4 ⊢ (𝐸‘ndx) = 𝑁 |
13 | 12 | opeq1i 3647 | . . 3 ⊢ 〈(𝐸‘ndx), + 〉 = 〈𝑁, + 〉 |
14 | 11, 13, 4 | 3eltr4g 2180 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 〈(𝐸‘ndx), + 〉 ∈ 𝐺) |
15 | 3, 6, 7, 14 | opelstrsl 11739 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → + = (𝐸‘𝐺)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1296 ∈ wcel 1445 Vcvv 2633 {cpr 3467 〈cop 3469 class class class wbr 3867 ‘cfv 5049 < clt 7619 ℕcn 8520 ndxcnx 11640 Slot cslot 11642 Basecbs 11643 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-addcom 7542 ax-addass 7544 ax-distr 7546 ax-i2m1 7547 ax-0lt1 7548 ax-0id 7550 ax-rnegex 7551 ax-cnre 7553 ax-pre-ltirr 7554 ax-pre-ltwlin 7555 ax-pre-lttrn 7556 ax-pre-apti 7557 ax-pre-ltadd 7558 |
This theorem depends on definitions: df-bi 116 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-br 3868 df-opab 3922 df-mpt 3923 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624 df-le 7625 df-sub 7752 df-neg 7753 df-inn 8521 df-n0 8772 df-z 8849 df-uz 9119 df-fz 9574 df-struct 11645 df-ndx 11646 df-slot 11647 df-base 11649 |
This theorem is referenced by: (None) |
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