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| Mirrors > Home > MPE Home > Th. List > noseqrdg0 | Structured version Visualization version GIF version | ||
| Description: Initial value of a recursive definition generator on surreal sequences. (Contributed by Scott Fenton, 18-Apr-2025.) |
| Ref | Expression |
|---|---|
| om2noseq.1 | ⊢ (𝜑 → 𝐶 ∈ No ) |
| om2noseq.2 | ⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)) |
| om2noseq.3 | ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω)) |
| noseqrdg.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
| noseqrdg.2 | ⊢ (𝜑 → 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω)) |
| noseqrdg.3 | ⊢ (𝜑 → 𝑆 = ran 𝑅) |
| Ref | Expression |
|---|---|
| noseqrdg0 | ⊢ (𝜑 → (𝑆‘𝐶) = 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | om2noseq.1 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ No ) | |
| 2 | om2noseq.2 | . . . 4 ⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)) | |
| 3 | om2noseq.3 | . . . 4 ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω)) | |
| 4 | noseqrdg.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 5 | noseqrdg.2 | . . . 4 ⊢ (𝜑 → 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω)) | |
| 6 | noseqrdg.3 | . . . 4 ⊢ (𝜑 → 𝑆 = ran 𝑅) | |
| 7 | 1, 2, 3, 4, 5, 6 | noseqrdgfn 28312 | . . 3 ⊢ (𝜑 → 𝑆 Fn 𝑍) |
| 8 | 7 | fnfund 6669 | . 2 ⊢ (𝜑 → Fun 𝑆) |
| 9 | frfnom 8475 | . . . . 5 ⊢ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω) Fn ω | |
| 10 | 5 | fneq1d 6661 | . . . . 5 ⊢ (𝜑 → (𝑅 Fn ω ↔ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω) Fn ω)) |
| 11 | 9, 10 | mpbiri 258 | . . . 4 ⊢ (𝜑 → 𝑅 Fn ω) |
| 12 | peano1 7910 | . . . 4 ⊢ ∅ ∈ ω | |
| 13 | fnfvelrn 7100 | . . . 4 ⊢ ((𝑅 Fn ω ∧ ∅ ∈ ω) → (𝑅‘∅) ∈ ran 𝑅) | |
| 14 | 11, 12, 13 | sylancl 586 | . . 3 ⊢ (𝜑 → (𝑅‘∅) ∈ ran 𝑅) |
| 15 | 5 | fveq1d 6908 | . . . 4 ⊢ (𝜑 → (𝑅‘∅) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω)‘∅)) |
| 16 | opex 5469 | . . . . 5 ⊢ 〈𝐶, 𝐴〉 ∈ V | |
| 17 | fr0g 8476 | . . . . 5 ⊢ (〈𝐶, 𝐴〉 ∈ V → ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω)‘∅) = 〈𝐶, 𝐴〉) | |
| 18 | 16, 17 | ax-mp 5 | . . . 4 ⊢ ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω)‘∅) = 〈𝐶, 𝐴〉 |
| 19 | 15, 18 | eqtr2di 2794 | . . 3 ⊢ (𝜑 → 〈𝐶, 𝐴〉 = (𝑅‘∅)) |
| 20 | 14, 19, 6 | 3eltr4d 2856 | . 2 ⊢ (𝜑 → 〈𝐶, 𝐴〉 ∈ 𝑆) |
| 21 | funopfv 6958 | . 2 ⊢ (Fun 𝑆 → (〈𝐶, 𝐴〉 ∈ 𝑆 → (𝑆‘𝐶) = 𝐴)) | |
| 22 | 8, 20, 21 | sylc 65 | 1 ⊢ (𝜑 → (𝑆‘𝐶) = 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 〈cop 4632 ↦ cmpt 5225 ran crn 5686 ↾ cres 5687 “ cima 5688 Fun wfun 6555 Fn wfn 6556 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ωcom 7887 reccrdg 8449 No csur 27684 1s c1s 27868 +s cadds 27992 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-ot 4635 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-nadd 8704 df-no 27687 df-slt 27688 df-bday 27689 df-sle 27790 df-sslt 27826 df-scut 27828 df-0s 27869 df-1s 27870 df-made 27886 df-old 27887 df-left 27889 df-right 27890 df-norec2 27982 df-adds 27993 |
| This theorem is referenced by: seqs1 28316 |
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