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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 8496 | . . 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 3499 | . . . 4 ⊢ (𝐾 ∈ ω → 𝐾 ∈ V) | |
5 | 4 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ ω) → 𝐾 ∈ V) |
6 | fvex 6920 | . . 3 ⊢ (𝐻‘𝐾) ∈ V | |
7 | simpl 482 | . . . . . . . 8 ⊢ ((𝑢 = 𝐾 ∧ 𝑣 = (𝐻‘𝐾)) → 𝑢 = 𝐾) | |
8 | 7 | fveq2d 6911 | . . . . . . 7 ⊢ ((𝑢 = 𝐾 ∧ 𝑣 = (𝐻‘𝐾)) → (𝐺‘𝑢) = (𝐺‘𝐾)) |
9 | 8 | oveq2d 7447 | . . . . . 6 ⊢ ((𝑢 = 𝐾 ∧ 𝑣 = (𝐻‘𝐾)) → (𝐴 ↑o (𝐺‘𝑢)) = (𝐴 ↑o (𝐺‘𝐾))) |
10 | 8 | fveq2d 6911 | . . . . . 6 ⊢ ((𝑢 = 𝐾 ∧ 𝑣 = (𝐻‘𝐾)) → (𝐹‘(𝐺‘𝑢)) = (𝐹‘(𝐺‘𝐾))) |
11 | 9, 10 | oveq12d 7449 | . . . . 5 ⊢ ((𝑢 = 𝐾 ∧ 𝑣 = (𝐻‘𝐾)) → ((𝐴 ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) = ((𝐴 ↑o (𝐺‘𝐾)) ·o (𝐹‘(𝐺‘𝐾)))) |
12 | simpr 484 | . . . . 5 ⊢ ((𝑢 = 𝐾 ∧ 𝑣 = (𝐻‘𝐾)) → 𝑣 = (𝐻‘𝐾)) | |
13 | 11, 12 | oveq12d 7449 | . . . 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 6907 | . . . . . . . 8 ⊢ (𝑘 = 𝑢 → (𝐺‘𝑘) = (𝐺‘𝑢)) | |
15 | 14 | oveq2d 7447 | . . . . . . 7 ⊢ (𝑘 = 𝑢 → (𝐴 ↑o (𝐺‘𝑘)) = (𝐴 ↑o (𝐺‘𝑢))) |
16 | 14 | fveq2d 6911 | . . . . . . 7 ⊢ (𝑘 = 𝑢 → (𝐹‘(𝐺‘𝑘)) = (𝐹‘(𝐺‘𝑢))) |
17 | 15, 16 | oveq12d 7449 | . . . . . 6 ⊢ (𝑘 = 𝑢 → ((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) = ((𝐴 ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢)))) |
18 | 17 | oveq1d 7446 | . . . . 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 7439 | . . . . 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 | cbvmpov 7528 | . . . 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 7464 | . . . 4 ⊢ (((𝐴 ↑o (𝐺‘𝐾)) ·o (𝐹‘(𝐺‘𝐾))) +o (𝐻‘𝐾)) ∈ V | |
22 | 13, 20, 21 | ovmpoa 7588 | . . 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 586 | . 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 2775 | 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 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 E cep 5588 dom cdm 5689 Oncon0 6386 suc csuc 6388 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 ωcom 7887 supp csupp 8184 seqωcseqom 8486 +o coa 8502 ·o comu 8503 ↑o coe 8504 OrdIsocoi 9547 CNF ccnf 9699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-seqom 8487 |
This theorem is referenced by: cantnfle 9709 cantnflt 9710 cantnfp1lem3 9718 cantnflem1d 9726 cantnflem1 9727 cnfcomlem 9737 |
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