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Mirrors > Home > MPE Home > Th. List > Mathboxes > breprexplemb | Structured version Visualization version GIF version |
Description: Lemma for breprexp 33476 (closure). (Contributed by Thierry Arnoux, 7-Dec-2021.) |
Ref | Expression |
---|---|
breprexp.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
breprexp.s | ⊢ (𝜑 → 𝑆 ∈ ℕ0) |
breprexp.z | ⊢ (𝜑 → 𝑍 ∈ ℂ) |
breprexp.h | ⊢ (𝜑 → 𝐿:(0..^𝑆)⟶(ℂ ↑m ℕ)) |
breprexplemb.x | ⊢ (𝜑 → 𝑋 ∈ (0..^𝑆)) |
breprexplemb.y | ⊢ (𝜑 → 𝑌 ∈ ℕ) |
Ref | Expression |
---|---|
breprexplemb | ⊢ (𝜑 → ((𝐿‘𝑋)‘𝑌) ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breprexp.h | . . . 4 ⊢ (𝜑 → 𝐿:(0..^𝑆)⟶(ℂ ↑m ℕ)) | |
2 | breprexplemb.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (0..^𝑆)) | |
3 | 1, 2 | ffvelcdmd 7072 | . . 3 ⊢ (𝜑 → (𝐿‘𝑋) ∈ (ℂ ↑m ℕ)) |
4 | cnex 11173 | . . . 4 ⊢ ℂ ∈ V | |
5 | nnex 12200 | . . . 4 ⊢ ℕ ∈ V | |
6 | 4, 5 | elmap 8848 | . . 3 ⊢ ((𝐿‘𝑋) ∈ (ℂ ↑m ℕ) ↔ (𝐿‘𝑋):ℕ⟶ℂ) |
7 | 3, 6 | sylib 217 | . 2 ⊢ (𝜑 → (𝐿‘𝑋):ℕ⟶ℂ) |
8 | breprexplemb.y | . 2 ⊢ (𝜑 → 𝑌 ∈ ℕ) | |
9 | 7, 8 | ffvelcdmd 7072 | 1 ⊢ (𝜑 → ((𝐿‘𝑋)‘𝑌) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ⟶wf 6528 ‘cfv 6532 (class class class)co 7393 ↑m cmap 8803 ℂcc 11090 0cc0 11092 ℕcn 12194 ℕ0cn0 12454 ..^cfzo 13609 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-1cn 11150 ax-addcl 11152 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-map 8805 df-nn 12195 |
This theorem is referenced by: breprexplemc 33475 circlemeth 33483 |
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