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Theorem seqsp1 28332
Description: The value of the surreal sequence builder at a successor. (Contributed by Scott Fenton, 19-Apr-2025.)
Hypotheses
Ref Expression
seqsp1.1 (𝜑𝑀 No )
seqsp1.2 (𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝑀) “ ω))
seqsp1.3 (𝜑𝑁𝑍)
Assertion
Ref Expression
seqsp1 (𝜑 → (seqs𝑀( + , 𝐹)‘(𝑁 +s 1s )) = ((seqs𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 +s 1s ))))

Proof of Theorem seqsp1
Dummy variables 𝑦 𝑧 𝑤 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 seqsp1.3 . . 3 (𝜑𝑁𝑍)
2 seqsp1.1 . . . 4 (𝜑𝑀 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 (𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝑀) “ ω))
5 oveq1 7438 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 +s 1s ) = (𝑦 +s 1s ))
65cbvmptv 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))
86, 7ax-mp 5 . . . . . 6 rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝑀) = rec((𝑦 ∈ V ↦ (𝑦 +s 1s )), 𝑀)
98imaeq1i 6077 . . . . 5 (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝑀) “ ω) = (rec((𝑦 ∈ V ↦ (𝑦 +s 1s )), 𝑀) “ ω)
104, 9eqtrdi 2791 . . . 4 (𝜑𝑍 = (rec((𝑦 ∈ V ↦ (𝑦 +s 1s )), 𝑀) “ ω))
11 fvexd 6922 . . . 4 (𝜑 → (𝐹𝑀) ∈ 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, (𝐹𝑀)⟩) ↾ ω) = (rec((𝑦 ∈ V, 𝑧 ∈ V ↦ ⟨(𝑦 +s 1s ), (𝑦(𝑤 ∈ V, 𝑡 ∈ V ↦ (𝑡 + (𝐹‘(𝑤 +s 1s ))))𝑧)⟩), ⟨𝑀, (𝐹𝑀)⟩) ↾ ω))
1312seqsval 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, (𝐹𝑀)⟩) ↾ ω))
142, 3, 10, 11, 12, 13noseqrdgsuc 28329 . . 3 ((𝜑𝑁𝑍) → (seqs𝑀( + , 𝐹)‘(𝑁 +s 1s )) = (𝑁(𝑤 ∈ V, 𝑡 ∈ V ↦ (𝑡 + (𝐹‘(𝑤 +s 1s ))))(seqs𝑀( + , 𝐹)‘𝑁)))
151, 14mpdan 687 . 2 (𝜑 → (seqs𝑀( + , 𝐹)‘(𝑁 +s 1s )) = (𝑁(𝑤 ∈ V, 𝑡 ∈ V ↦ (𝑡 + (𝐹‘(𝑤 +s 1s ))))(seqs𝑀( + , 𝐹)‘𝑁)))
161elexd 3502 . . 3 (𝜑𝑁 ∈ V)
17 fvex 6920 . . 3 (seqs𝑀( + , 𝐹)‘𝑁) ∈ V
18 fvoveq1 7454 . . . . 5 (𝑤 = 𝑁 → (𝐹‘(𝑤 +s 1s )) = (𝐹‘(𝑁 +s 1s )))
1918oveq2d 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
2319, 20, 21, 22ovmpo 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 ))))
2416, 17, 23sylancl 586 . 2 (𝜑 → (𝑁(𝑤 ∈ V, 𝑡 ∈ V ↦ (𝑡 + (𝐹‘(𝑤 +s 1s ))))(seqs𝑀( + , 𝐹)‘𝑁)) = ((seqs𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 +s 1s ))))
2515, 24eqtrd 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
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