![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 28327 | . . 3 ⊢ (𝜑 → 𝑆 Fn 𝑍) |
8 | 7 | fnfund 6670 | . 2 ⊢ (𝜑 → Fun 𝑆) |
9 | frfnom 8474 | . . . . 5 ⊢ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω) Fn ω | |
10 | 5 | fneq1d 6662 | . . . . 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 7911 | . . . 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 6909 | . . . 4 ⊢ (𝜑 → (𝑅‘∅) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω)‘∅)) |
16 | opex 5475 | . . . . 5 ⊢ 〈𝐶, 𝐴〉 ∈ V | |
17 | fr0g 8475 | . . . . 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 2792 | . . 3 ⊢ (𝜑 → 〈𝐶, 𝐴〉 = (𝑅‘∅)) |
20 | 14, 19, 6 | 3eltr4d 2854 | . 2 ⊢ (𝜑 → 〈𝐶, 𝐴〉 ∈ 𝑆) |
21 | funopfv 6959 | . 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 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 〈cop 4637 ↦ cmpt 5231 ran crn 5690 ↾ cres 5691 “ cima 5692 Fun wfun 6557 Fn wfn 6558 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 ωcom 7887 reccrdg 8448 No csur 27699 1s c1s 27883 +s cadds 28007 |
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-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 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-rmo 3378 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-tp 4636 df-op 4638 df-ot 4640 df-uni 4913 df-int 4952 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-se 5642 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-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-nadd 8703 df-no 27702 df-slt 27703 df-bday 27704 df-sle 27805 df-sslt 27841 df-scut 27843 df-0s 27884 df-1s 27885 df-made 27901 df-old 27902 df-left 27904 df-right 27905 df-norec2 27997 df-adds 28008 |
This theorem is referenced by: seqs1 28331 |
Copyright terms: Public domain | W3C validator |