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Mirrors > Home > ILE Home > Th. List > ialgrlemconst | GIF version |
Description: Lemma for ialgr0 11714. Closure of a constant function, in a form suitable for theorems such as seq3-1 10226 or seqf 10227. (Contributed by Jim Kingdon, 22-Jul-2021.) |
Ref | Expression |
---|---|
ialgrlemconst.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
ialgrlemconst.a | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
Ref | Expression |
---|---|
ialgrlemconst | ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ((𝑍 × {𝐴})‘𝑥) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ialgrlemconst.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
2 | ialgrlemconst.z | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
3 | 2 | eleq2i 2204 | . . . 4 ⊢ (𝑥 ∈ 𝑍 ↔ 𝑥 ∈ (ℤ≥‘𝑀)) |
4 | 3 | biimpri 132 | . . 3 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → 𝑥 ∈ 𝑍) |
5 | fvconst2g 5627 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑥 ∈ 𝑍) → ((𝑍 × {𝐴})‘𝑥) = 𝐴) | |
6 | 1, 4, 5 | syl2an 287 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ((𝑍 × {𝐴})‘𝑥) = 𝐴) |
7 | 1 | adantr 274 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → 𝐴 ∈ 𝑆) |
8 | 6, 7 | eqeltrd 2214 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ((𝑍 × {𝐴})‘𝑥) ∈ 𝑆) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 {csn 3522 × cxp 4532 ‘cfv 5118 ℤ≥cuz 9319 |
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-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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 |
This theorem is referenced by: ialgr0 11714 algrp1 11716 |
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