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Mirrors > Home > MPE Home > Th. List > Mathboxes > breprexplemb | Structured version Visualization version GIF version |
Description: Lemma for breprexp 32613 (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 | ffvelrnd 6962 | . . 3 ⊢ (𝜑 → (𝐿‘𝑋) ∈ (ℂ ↑m ℕ)) |
4 | cnex 10952 | . . . 4 ⊢ ℂ ∈ V | |
5 | nnex 11979 | . . . 4 ⊢ ℕ ∈ V | |
6 | 4, 5 | elmap 8659 | . . 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 | ffvelrnd 6962 | 1 ⊢ (𝜑 → ((𝐿‘𝑋)‘𝑌) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ↑m cmap 8615 ℂcc 10869 0cc0 10871 ℕcn 11973 ℕ0cn0 12233 ..^cfzo 13382 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-1cn 10929 ax-addcl 10931 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-map 8617 df-nn 11974 |
This theorem is referenced by: breprexplemc 32612 circlemeth 32620 |
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