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Mirrors > Home > MPE Home > Th. List > Mathboxes > breprexplemb | Structured version Visualization version GIF version |
Description: Lemma for breprexp 30711 (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 6360 | . . 3 ⊢ (𝜑 → (𝐿‘𝑋) ∈ (ℂ ↑𝑚 ℕ)) |
4 | cnex 10017 | . . . 4 ⊢ ℂ ∈ V | |
5 | nnex 11026 | . . . 4 ⊢ ℕ ∈ V | |
6 | 4, 5 | elmap 7886 | . . 3 ⊢ ((𝐿‘𝑋) ∈ (ℂ ↑𝑚 ℕ) ↔ (𝐿‘𝑋):ℕ⟶ℂ) |
7 | 3, 6 | sylib 208 | . 2 ⊢ (𝜑 → (𝐿‘𝑋):ℕ⟶ℂ) |
8 | breprexplemb.y | . 2 ⊢ (𝜑 → 𝑌 ∈ ℕ) | |
9 | 7, 8 | ffvelrnd 6360 | 1 ⊢ (𝜑 → ((𝐿‘𝑋)‘𝑌) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 ℂcc 9934 0cc0 9936 ℕcn 11020 ℕ0cn0 11292 ..^cfzo 12465 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-i2m1 10004 ax-1ne0 10005 ax-rrecex 10008 ax-cnre 10009 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-map 7859 df-nn 11021 |
This theorem is referenced by: breprexplemc 30710 circlemeth 30718 |
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