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| Mirrors > Home > MPE Home > Th. List > noseqrdglem | Structured version Visualization version GIF version | ||
| Description: A helper lemma for the value of a recursive defintion 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)) | 
| Ref | Expression | 
|---|---|
| noseqrdglem | ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → 〈𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉 ∈ ran 𝑅) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | om2noseq.1 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ No ) | |
| 2 | om2noseq.2 | . . . . . 6 ⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)) | |
| 3 | om2noseq.3 | . . . . . 6 ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω)) | |
| 4 | 1, 2, 3 | om2noseqf1o 28307 | . . . . 5 ⊢ (𝜑 → 𝐺:ω–1-1-onto→𝑍) | 
| 5 | f1ocnvdm 7305 | . . . . 5 ⊢ ((𝐺:ω–1-1-onto→𝑍 ∧ 𝐵 ∈ 𝑍) → (◡𝐺‘𝐵) ∈ ω) | |
| 6 | 4, 5 | sylan 580 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (◡𝐺‘𝐵) ∈ ω) | 
| 7 | noseqrdg.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 8 | noseqrdg.2 | . . . . 5 ⊢ (𝜑 → 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω)) | |
| 9 | 1, 2, 3, 7, 8 | om2noseqrdg 28310 | . . . 4 ⊢ ((𝜑 ∧ (◡𝐺‘𝐵) ∈ ω) → (𝑅‘(◡𝐺‘𝐵)) = 〈(𝐺‘(◡𝐺‘𝐵)), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉) | 
| 10 | 6, 9 | syldan 591 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝑅‘(◡𝐺‘𝐵)) = 〈(𝐺‘(◡𝐺‘𝐵)), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉) | 
| 11 | f1ocnvfv2 7297 | . . . . 5 ⊢ ((𝐺:ω–1-1-onto→𝑍 ∧ 𝐵 ∈ 𝑍) → (𝐺‘(◡𝐺‘𝐵)) = 𝐵) | |
| 12 | 4, 11 | sylan 580 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝐺‘(◡𝐺‘𝐵)) = 𝐵) | 
| 13 | 12 | opeq1d 4879 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → 〈(𝐺‘(◡𝐺‘𝐵)), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉 = 〈𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉) | 
| 14 | 10, 13 | eqtrd 2777 | . 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝑅‘(◡𝐺‘𝐵)) = 〈𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉) | 
| 15 | frfnom 8475 | . . . . 5 ⊢ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω) Fn ω | |
| 16 | 8 | fneq1d 6661 | . . . . 5 ⊢ (𝜑 → (𝑅 Fn ω ↔ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω) Fn ω)) | 
| 17 | 15, 16 | mpbiri 258 | . . . 4 ⊢ (𝜑 → 𝑅 Fn ω) | 
| 18 | 17 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → 𝑅 Fn ω) | 
| 19 | 18, 6 | fnfvelrnd 7102 | . 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝑅‘(◡𝐺‘𝐵)) ∈ ran 𝑅) | 
| 20 | 14, 19 | eqeltrrd 2842 | 1 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → 〈𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉 ∈ ran 𝑅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 ↦ cmpt 5225 ◡ccnv 5684 ran crn 5686 ↾ cres 5687 “ cima 5688 Fn wfn 6556 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ωcom 7887 2nd c2nd 8013 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: noseqrdgfn 28312 noseqrdgsuc 28314 | 
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