![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > seqsp1 | Structured version Visualization version GIF version |
Description: The value of the surreal sequence builder at a successor. (Contributed by Scott Fenton, 19-Apr-2025.) |
Ref | Expression |
---|---|
seqsp1.1 | ⊢ (𝜑 → 𝑀 ∈ No ) |
seqsp1.2 | ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝑀) “ ω)) |
seqsp1.3 | ⊢ (𝜑 → 𝑁 ∈ 𝑍) |
Ref | Expression |
---|---|
seqsp1 | ⊢ (𝜑 → (seqs𝑀( + , 𝐹)‘(𝑁 +s 1s )) = ((seqs𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 +s 1s )))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seqsp1.3 | . . 3 ⊢ (𝜑 → 𝑁 ∈ 𝑍) | |
2 | seqsp1.1 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ No ) | |
3 | eqidd 2736 | . . . 4 ⊢ (𝜑 → (rec((𝑦 ∈ V ↦ (𝑦 +s 1s )), 𝑀) ↾ ω) = (rec((𝑦 ∈ V ↦ (𝑦 +s 1s )), 𝑀) ↾ ω)) | |
4 | seqsp1.2 | . . . . 5 ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝑀) “ ω)) | |
5 | oveq1 7438 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 +s 1s ) = (𝑦 +s 1s )) | |
6 | 5 | cbvmptv 5261 | . . . . . . 7 ⊢ (𝑥 ∈ V ↦ (𝑥 +s 1s )) = (𝑦 ∈ V ↦ (𝑦 +s 1s )) |
7 | rdgeq1 8450 | . . . . . . 7 ⊢ ((𝑥 ∈ V ↦ (𝑥 +s 1s )) = (𝑦 ∈ V ↦ (𝑦 +s 1s )) → rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝑀) = rec((𝑦 ∈ V ↦ (𝑦 +s 1s )), 𝑀)) | |
8 | 6, 7 | ax-mp 5 | . . . . . 6 ⊢ rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝑀) = rec((𝑦 ∈ V ↦ (𝑦 +s 1s )), 𝑀) |
9 | 8 | imaeq1i 6077 | . . . . 5 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝑀) “ ω) = (rec((𝑦 ∈ V ↦ (𝑦 +s 1s )), 𝑀) “ ω) |
10 | 4, 9 | eqtrdi 2791 | . . . 4 ⊢ (𝜑 → 𝑍 = (rec((𝑦 ∈ V ↦ (𝑦 +s 1s )), 𝑀) “ ω)) |
11 | fvexd 6922 | . . . 4 ⊢ (𝜑 → (𝐹‘𝑀) ∈ V) | |
12 | eqidd 2736 | . . . 4 ⊢ (𝜑 → (rec((𝑦 ∈ V, 𝑧 ∈ V ↦ 〈(𝑦 +s 1s ), (𝑦(𝑤 ∈ V, 𝑡 ∈ V ↦ (𝑡 + (𝐹‘(𝑤 +s 1s ))))𝑧)〉), 〈𝑀, (𝐹‘𝑀)〉) ↾ ω) = (rec((𝑦 ∈ V, 𝑧 ∈ V ↦ 〈(𝑦 +s 1s ), (𝑦(𝑤 ∈ V, 𝑡 ∈ V ↦ (𝑡 + (𝐹‘(𝑤 +s 1s ))))𝑧)〉), 〈𝑀, (𝐹‘𝑀)〉) ↾ ω)) | |
13 | 12 | seqsval 28309 | . . . 4 ⊢ (𝜑 → seqs𝑀( + , 𝐹) = ran (rec((𝑦 ∈ V, 𝑧 ∈ V ↦ 〈(𝑦 +s 1s ), (𝑦(𝑤 ∈ V, 𝑡 ∈ V ↦ (𝑡 + (𝐹‘(𝑤 +s 1s ))))𝑧)〉), 〈𝑀, (𝐹‘𝑀)〉) ↾ ω)) |
14 | 2, 3, 10, 11, 12, 13 | noseqrdgsuc 28329 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍) → (seqs𝑀( + , 𝐹)‘(𝑁 +s 1s )) = (𝑁(𝑤 ∈ V, 𝑡 ∈ V ↦ (𝑡 + (𝐹‘(𝑤 +s 1s ))))(seqs𝑀( + , 𝐹)‘𝑁))) |
15 | 1, 14 | mpdan 687 | . 2 ⊢ (𝜑 → (seqs𝑀( + , 𝐹)‘(𝑁 +s 1s )) = (𝑁(𝑤 ∈ V, 𝑡 ∈ V ↦ (𝑡 + (𝐹‘(𝑤 +s 1s ))))(seqs𝑀( + , 𝐹)‘𝑁))) |
16 | 1 | elexd 3502 | . . 3 ⊢ (𝜑 → 𝑁 ∈ V) |
17 | fvex 6920 | . . 3 ⊢ (seqs𝑀( + , 𝐹)‘𝑁) ∈ V | |
18 | fvoveq1 7454 | . . . . 5 ⊢ (𝑤 = 𝑁 → (𝐹‘(𝑤 +s 1s )) = (𝐹‘(𝑁 +s 1s ))) | |
19 | 18 | oveq2d 7447 | . . . 4 ⊢ (𝑤 = 𝑁 → (𝑡 + (𝐹‘(𝑤 +s 1s ))) = (𝑡 + (𝐹‘(𝑁 +s 1s )))) |
20 | oveq1 7438 | . . . 4 ⊢ (𝑡 = (seqs𝑀( + , 𝐹)‘𝑁) → (𝑡 + (𝐹‘(𝑁 +s 1s ))) = ((seqs𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 +s 1s )))) | |
21 | eqid 2735 | . . . 4 ⊢ (𝑤 ∈ V, 𝑡 ∈ V ↦ (𝑡 + (𝐹‘(𝑤 +s 1s )))) = (𝑤 ∈ V, 𝑡 ∈ V ↦ (𝑡 + (𝐹‘(𝑤 +s 1s )))) | |
22 | ovex 7464 | . . . 4 ⊢ ((seqs𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 +s 1s ))) ∈ V | |
23 | 19, 20, 21, 22 | ovmpo 7593 | . . 3 ⊢ ((𝑁 ∈ V ∧ (seqs𝑀( + , 𝐹)‘𝑁) ∈ V) → (𝑁(𝑤 ∈ V, 𝑡 ∈ V ↦ (𝑡 + (𝐹‘(𝑤 +s 1s ))))(seqs𝑀( + , 𝐹)‘𝑁)) = ((seqs𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 +s 1s )))) |
24 | 16, 17, 23 | sylancl 586 | . 2 ⊢ (𝜑 → (𝑁(𝑤 ∈ V, 𝑡 ∈ V ↦ (𝑡 + (𝐹‘(𝑤 +s 1s ))))(seqs𝑀( + , 𝐹)‘𝑁)) = ((seqs𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 +s 1s )))) |
25 | 15, 24 | eqtrd 2775 | 1 ⊢ (𝜑 → (seqs𝑀( + , 𝐹)‘(𝑁 +s 1s )) = ((seqs𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 +s 1s )))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 〈cop 4637 ↦ cmpt 5231 ↾ cres 5691 “ cima 5692 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 ωcom 7887 reccrdg 8448 No csur 27699 1s c1s 27883 +s cadds 28007 seqscseqs 28304 |
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 df-seqs 28305 |
This theorem is referenced by: expsp1 28427 |
Copyright terms: Public domain | W3C validator |