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Mirrors > Home > ILE Home > Th. List > ialgrlemconst | GIF version |
Description: Lemma for ialgr0 12038. Closure of a constant function, in a form suitable for theorems such as seq3-1 10457 or seqf 10458. (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 2244 | . . . 4 ⊢ (𝑥 ∈ 𝑍 ↔ 𝑥 ∈ (ℤ≥‘𝑀)) |
4 | 3 | biimpri 133 | . . 3 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → 𝑥 ∈ 𝑍) |
5 | fvconst2g 5730 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑥 ∈ 𝑍) → ((𝑍 × {𝐴})‘𝑥) = 𝐴) | |
6 | 1, 4, 5 | syl2an 289 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ((𝑍 × {𝐴})‘𝑥) = 𝐴) |
7 | 1 | adantr 276 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → 𝐴 ∈ 𝑆) |
8 | 6, 7 | eqeltrd 2254 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ((𝑍 × {𝐴})‘𝑥) ∈ 𝑆) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 {csn 3592 × cxp 4624 ‘cfv 5216 ℤ≥cuz 9526 |
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-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-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 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-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-fv 5224 |
This theorem is referenced by: ialgr0 12038 algrp1 12040 mulgnn0z 12963 mulgnndir 12965 mulgpropdg 12978 |
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