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Mirrors > Home > MPE Home > Th. List > cantnfsuc | Structured version Visualization version GIF version |
Description: The value of the recursive function 𝐻 at a successor. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.) |
Ref | Expression |
---|---|
cantnfs.s | ⊢ 𝑆 = dom (𝐴 CNF 𝐵) |
cantnfs.a | ⊢ (𝜑 → 𝐴 ∈ On) |
cantnfs.b | ⊢ (𝜑 → 𝐵 ∈ On) |
cantnfcl.g | ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) |
cantnfcl.f | ⊢ (𝜑 → 𝐹 ∈ 𝑆) |
cantnfval.h | ⊢ 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅) |
Ref | Expression |
---|---|
cantnfsuc | ⊢ ((𝜑 ∧ 𝐾 ∈ ω) → (𝐻‘suc 𝐾) = (((𝐴 ↑𝑜 (𝐺‘𝐾)) ·𝑜 (𝐹‘(𝐺‘𝐾))) +𝑜 (𝐻‘𝐾))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cantnfval.h | . . . 4 ⊢ 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅) | |
2 | 1 | seqomsuc 7439 | . . 3 ⊢ (𝐾 ∈ ω → (𝐻‘suc 𝐾) = (𝐾(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧))(𝐻‘𝐾))) |
3 | 2 | adantl 481 | . 2 ⊢ ((𝜑 ∧ 𝐾 ∈ ω) → (𝐻‘suc 𝐾) = (𝐾(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧))(𝐻‘𝐾))) |
4 | elex 3185 | . . . 4 ⊢ (𝐾 ∈ ω → 𝐾 ∈ V) | |
5 | 4 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ ω) → 𝐾 ∈ V) |
6 | fvex 6113 | . . 3 ⊢ (𝐻‘𝐾) ∈ V | |
7 | simpl 472 | . . . . . . . 8 ⊢ ((𝑢 = 𝐾 ∧ 𝑣 = (𝐻‘𝐾)) → 𝑢 = 𝐾) | |
8 | 7 | fveq2d 6107 | . . . . . . 7 ⊢ ((𝑢 = 𝐾 ∧ 𝑣 = (𝐻‘𝐾)) → (𝐺‘𝑢) = (𝐺‘𝐾)) |
9 | 8 | oveq2d 6565 | . . . . . 6 ⊢ ((𝑢 = 𝐾 ∧ 𝑣 = (𝐻‘𝐾)) → (𝐴 ↑𝑜 (𝐺‘𝑢)) = (𝐴 ↑𝑜 (𝐺‘𝐾))) |
10 | 8 | fveq2d 6107 | . . . . . 6 ⊢ ((𝑢 = 𝐾 ∧ 𝑣 = (𝐻‘𝐾)) → (𝐹‘(𝐺‘𝑢)) = (𝐹‘(𝐺‘𝐾))) |
11 | 9, 10 | oveq12d 6567 | . . . . 5 ⊢ ((𝑢 = 𝐾 ∧ 𝑣 = (𝐻‘𝐾)) → ((𝐴 ↑𝑜 (𝐺‘𝑢)) ·𝑜 (𝐹‘(𝐺‘𝑢))) = ((𝐴 ↑𝑜 (𝐺‘𝐾)) ·𝑜 (𝐹‘(𝐺‘𝐾)))) |
12 | simpr 476 | . . . . 5 ⊢ ((𝑢 = 𝐾 ∧ 𝑣 = (𝐻‘𝐾)) → 𝑣 = (𝐻‘𝐾)) | |
13 | 11, 12 | oveq12d 6567 | . . . 4 ⊢ ((𝑢 = 𝐾 ∧ 𝑣 = (𝐻‘𝐾)) → (((𝐴 ↑𝑜 (𝐺‘𝑢)) ·𝑜 (𝐹‘(𝐺‘𝑢))) +𝑜 𝑣) = (((𝐴 ↑𝑜 (𝐺‘𝐾)) ·𝑜 (𝐹‘(𝐺‘𝐾))) +𝑜 (𝐻‘𝐾))) |
14 | fveq2 6103 | . . . . . . . 8 ⊢ (𝑘 = 𝑢 → (𝐺‘𝑘) = (𝐺‘𝑢)) | |
15 | 14 | oveq2d 6565 | . . . . . . 7 ⊢ (𝑘 = 𝑢 → (𝐴 ↑𝑜 (𝐺‘𝑘)) = (𝐴 ↑𝑜 (𝐺‘𝑢))) |
16 | 14 | fveq2d 6107 | . . . . . . 7 ⊢ (𝑘 = 𝑢 → (𝐹‘(𝐺‘𝑘)) = (𝐹‘(𝐺‘𝑢))) |
17 | 15, 16 | oveq12d 6567 | . . . . . 6 ⊢ (𝑘 = 𝑢 → ((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) = ((𝐴 ↑𝑜 (𝐺‘𝑢)) ·𝑜 (𝐹‘(𝐺‘𝑢)))) |
18 | 17 | oveq1d 6564 | . . . . 5 ⊢ (𝑘 = 𝑢 → (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧) = (((𝐴 ↑𝑜 (𝐺‘𝑢)) ·𝑜 (𝐹‘(𝐺‘𝑢))) +𝑜 𝑧)) |
19 | oveq2 6557 | . . . . 5 ⊢ (𝑧 = 𝑣 → (((𝐴 ↑𝑜 (𝐺‘𝑢)) ·𝑜 (𝐹‘(𝐺‘𝑢))) +𝑜 𝑧) = (((𝐴 ↑𝑜 (𝐺‘𝑢)) ·𝑜 (𝐹‘(𝐺‘𝑢))) +𝑜 𝑣)) | |
20 | 18, 19 | cbvmpt2v 6633 | . . . 4 ⊢ (𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)) = (𝑢 ∈ V, 𝑣 ∈ V ↦ (((𝐴 ↑𝑜 (𝐺‘𝑢)) ·𝑜 (𝐹‘(𝐺‘𝑢))) +𝑜 𝑣)) |
21 | ovex 6577 | . . . 4 ⊢ (((𝐴 ↑𝑜 (𝐺‘𝐾)) ·𝑜 (𝐹‘(𝐺‘𝐾))) +𝑜 (𝐻‘𝐾)) ∈ V | |
22 | 13, 20, 21 | ovmpt2a 6689 | . . 3 ⊢ ((𝐾 ∈ V ∧ (𝐻‘𝐾) ∈ V) → (𝐾(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧))(𝐻‘𝐾)) = (((𝐴 ↑𝑜 (𝐺‘𝐾)) ·𝑜 (𝐹‘(𝐺‘𝐾))) +𝑜 (𝐻‘𝐾))) |
23 | 5, 6, 22 | sylancl 693 | . 2 ⊢ ((𝜑 ∧ 𝐾 ∈ ω) → (𝐾(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧))(𝐻‘𝐾)) = (((𝐴 ↑𝑜 (𝐺‘𝐾)) ·𝑜 (𝐹‘(𝐺‘𝐾))) +𝑜 (𝐻‘𝐾))) |
24 | 3, 23 | eqtrd 2644 | 1 ⊢ ((𝜑 ∧ 𝐾 ∈ ω) → (𝐻‘suc 𝐾) = (((𝐴 ↑𝑜 (𝐺‘𝐾)) ·𝑜 (𝐹‘(𝐺‘𝐾))) +𝑜 (𝐻‘𝐾))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 E cep 4947 dom cdm 5038 Oncon0 5640 suc csuc 5642 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ωcom 6957 supp csupp 7182 seq𝜔cseqom 7429 +𝑜 coa 7444 ·𝑜 comu 7445 ↑𝑜 coe 7446 OrdIsocoi 8297 CNF ccnf 8441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-seqom 7430 |
This theorem is referenced by: cantnfle 8451 cantnflt 8452 cantnfp1lem3 8460 cantnflem1d 8468 cantnflem1 8469 cnfcomlem 8479 |
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