MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  noseqrdgsuc Structured version   Visualization version   GIF version

Theorem noseqrdgsuc 28314
Description: Successor value of a recursive definition generator on surreal sequences. (Contributed by Scott Fenton, 19-Apr-2025.)
Hypotheses
Ref Expression
om2noseq.1 (𝜑𝐶 No )
om2noseq.2 (𝜑𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))
om2noseq.3 (𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))
noseqrdg.1 (𝜑𝐴𝑉)
noseqrdg.2 (𝜑𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))
noseqrdg.3 (𝜑𝑆 = ran 𝑅)
Assertion
Ref Expression
noseqrdgsuc ((𝜑𝐵𝑍) → (𝑆‘(𝐵 +s 1s )) = (𝐵𝐹(𝑆𝐵)))
Distinct variable groups:   𝑥,𝐶   𝑥,𝐹,𝑦   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑦)   𝐶(𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑍(𝑥,𝑦)

Proof of Theorem noseqrdgsuc
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 om2noseq.1 . . . . . . 7 (𝜑𝐶 No )
2 om2noseq.2 . . . . . . 7 (𝜑𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))
3 om2noseq.3 . . . . . . 7 (𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))
4 noseqrdg.1 . . . . . . 7 (𝜑𝐴𝑉)
5 noseqrdg.2 . . . . . . 7 (𝜑𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))
6 noseqrdg.3 . . . . . . 7 (𝜑𝑆 = ran 𝑅)
71, 2, 3, 4, 5, 6noseqrdgfn 28312 . . . . . 6 (𝜑𝑆 Fn 𝑍)
87adantr 480 . . . . 5 ((𝜑𝐵𝑍) → 𝑆 Fn 𝑍)
98fnfund 6669 . . . 4 ((𝜑𝐵𝑍) → Fun 𝑆)
103adantr 480 . . . . . . 7 ((𝜑𝐵𝑍) → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))
111adantr 480 . . . . . . 7 ((𝜑𝐵𝑍) → 𝐶 No )
12 simpr 484 . . . . . . 7 ((𝜑𝐵𝑍) → 𝐵𝑍)
1310, 11, 12noseqp1 28297 . . . . . 6 ((𝜑𝐵𝑍) → (𝐵 +s 1s ) ∈ 𝑍)
141, 2, 3, 4, 5noseqrdglem 28311 . . . . . 6 ((𝜑 ∧ (𝐵 +s 1s ) ∈ 𝑍) → ⟨(𝐵 +s 1s ), (2nd ‘(𝑅‘(𝐺‘(𝐵 +s 1s ))))⟩ ∈ ran 𝑅)
1513, 14syldan 591 . . . . 5 ((𝜑𝐵𝑍) → ⟨(𝐵 +s 1s ), (2nd ‘(𝑅‘(𝐺‘(𝐵 +s 1s ))))⟩ ∈ ran 𝑅)
166adantr 480 . . . . 5 ((𝜑𝐵𝑍) → 𝑆 = ran 𝑅)
1715, 16eleqtrrd 2844 . . . 4 ((𝜑𝐵𝑍) → ⟨(𝐵 +s 1s ), (2nd ‘(𝑅‘(𝐺‘(𝐵 +s 1s ))))⟩ ∈ 𝑆)
18 funopfv 6958 . . . 4 (Fun 𝑆 → (⟨(𝐵 +s 1s ), (2nd ‘(𝑅‘(𝐺‘(𝐵 +s 1s ))))⟩ ∈ 𝑆 → (𝑆‘(𝐵 +s 1s )) = (2nd ‘(𝑅‘(𝐺‘(𝐵 +s 1s ))))))
199, 17, 18sylc 65 . . 3 ((𝜑𝐵𝑍) → (𝑆‘(𝐵 +s 1s )) = (2nd ‘(𝑅‘(𝐺‘(𝐵 +s 1s )))))
201, 2, 3om2noseqf1o 28307 . . . . . . . 8 (𝜑𝐺:ω–1-1-onto𝑍)
2120adantr 480 . . . . . . 7 ((𝜑𝐵𝑍) → 𝐺:ω–1-1-onto𝑍)
22 f1ocnvdm 7305 . . . . . . . . 9 ((𝐺:ω–1-1-onto𝑍𝐵𝑍) → (𝐺𝐵) ∈ ω)
2320, 22sylan 580 . . . . . . . 8 ((𝜑𝐵𝑍) → (𝐺𝐵) ∈ ω)
24 peano2 7912 . . . . . . . 8 ((𝐺𝐵) ∈ ω → suc (𝐺𝐵) ∈ ω)
2523, 24syl 17 . . . . . . 7 ((𝜑𝐵𝑍) → suc (𝐺𝐵) ∈ ω)
2621, 25jca 511 . . . . . 6 ((𝜑𝐵𝑍) → (𝐺:ω–1-1-onto𝑍 ∧ suc (𝐺𝐵) ∈ ω))
272adantr 480 . . . . . . . 8 ((𝜑𝐵𝑍) → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))
2811, 27, 23om2noseqsuc 28303 . . . . . . 7 ((𝜑𝐵𝑍) → (𝐺‘suc (𝐺𝐵)) = ((𝐺‘(𝐺𝐵)) +s 1s ))
29 f1ocnvfv2 7297 . . . . . . . . 9 ((𝐺:ω–1-1-onto𝑍𝐵𝑍) → (𝐺‘(𝐺𝐵)) = 𝐵)
3020, 29sylan 580 . . . . . . . 8 ((𝜑𝐵𝑍) → (𝐺‘(𝐺𝐵)) = 𝐵)
3130oveq1d 7446 . . . . . . 7 ((𝜑𝐵𝑍) → ((𝐺‘(𝐺𝐵)) +s 1s ) = (𝐵 +s 1s ))
3228, 31eqtrd 2777 . . . . . 6 ((𝜑𝐵𝑍) → (𝐺‘suc (𝐺𝐵)) = (𝐵 +s 1s ))
33 f1ocnvfv 7298 . . . . . 6 ((𝐺:ω–1-1-onto𝑍 ∧ suc (𝐺𝐵) ∈ ω) → ((𝐺‘suc (𝐺𝐵)) = (𝐵 +s 1s ) → (𝐺‘(𝐵 +s 1s )) = suc (𝐺𝐵)))
3426, 32, 33sylc 65 . . . . 5 ((𝜑𝐵𝑍) → (𝐺‘(𝐵 +s 1s )) = suc (𝐺𝐵))
3534fveq2d 6910 . . . 4 ((𝜑𝐵𝑍) → (𝑅‘(𝐺‘(𝐵 +s 1s ))) = (𝑅‘suc (𝐺𝐵)))
3635fveq2d 6910 . . 3 ((𝜑𝐵𝑍) → (2nd ‘(𝑅‘(𝐺‘(𝐵 +s 1s )))) = (2nd ‘(𝑅‘suc (𝐺𝐵))))
3719, 36eqtrd 2777 . 2 ((𝜑𝐵𝑍) → (𝑆‘(𝐵 +s 1s )) = (2nd ‘(𝑅‘suc (𝐺𝐵))))
38 frsuc 8477 . . . . . . . . 9 ((𝐺𝐵) ∈ ω → ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc (𝐺𝐵)) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘(𝐺𝐵))))
3938adantl 481 . . . . . . . 8 ((𝜑 ∧ (𝐺𝐵) ∈ ω) → ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc (𝐺𝐵)) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘(𝐺𝐵))))
405fveq1d 6908 . . . . . . . . 9 (𝜑 → (𝑅‘suc (𝐺𝐵)) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc (𝐺𝐵)))
4140adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝐺𝐵) ∈ ω) → (𝑅‘suc (𝐺𝐵)) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc (𝐺𝐵)))
425fveq1d 6908 . . . . . . . . . 10 (𝜑 → (𝑅‘(𝐺𝐵)) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘(𝐺𝐵)))
4342fveq2d 6910 . . . . . . . . 9 (𝜑 → ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘(𝑅‘(𝐺𝐵))) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘(𝐺𝐵))))
4443adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝐺𝐵) ∈ ω) → ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘(𝑅‘(𝐺𝐵))) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘(𝐺𝐵))))
4539, 41, 443eqtr4d 2787 . . . . . . 7 ((𝜑 ∧ (𝐺𝐵) ∈ ω) → (𝑅‘suc (𝐺𝐵)) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘(𝑅‘(𝐺𝐵))))
461, 2, 3, 4, 5om2noseqrdg 28310 . . . . . . . . 9 ((𝜑 ∧ (𝐺𝐵) ∈ ω) → (𝑅‘(𝐺𝐵)) = ⟨(𝐺‘(𝐺𝐵)), (2nd ‘(𝑅‘(𝐺𝐵)))⟩)
4746fveq2d 6910 . . . . . . . 8 ((𝜑 ∧ (𝐺𝐵) ∈ ω) → ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘(𝑅‘(𝐺𝐵))) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘⟨(𝐺‘(𝐺𝐵)), (2nd ‘(𝑅‘(𝐺𝐵)))⟩))
48 df-ov 7434 . . . . . . . 8 ((𝐺‘(𝐺𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(𝐺𝐵)))) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘⟨(𝐺‘(𝐺𝐵)), (2nd ‘(𝑅‘(𝐺𝐵)))⟩)
4947, 48eqtr4di 2795 . . . . . . 7 ((𝜑 ∧ (𝐺𝐵) ∈ ω) → ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘(𝑅‘(𝐺𝐵))) = ((𝐺‘(𝐺𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(𝐺𝐵)))))
5045, 49eqtrd 2777 . . . . . 6 ((𝜑 ∧ (𝐺𝐵) ∈ ω) → (𝑅‘suc (𝐺𝐵)) = ((𝐺‘(𝐺𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(𝐺𝐵)))))
51 fvex 6919 . . . . . . 7 (𝐺‘(𝐺𝐵)) ∈ V
52 fvex 6919 . . . . . . 7 (2nd ‘(𝑅‘(𝐺𝐵))) ∈ V
53 oveq1 7438 . . . . . . . . 9 (𝑧 = (𝐺‘(𝐺𝐵)) → (𝑧 +s 1s ) = ((𝐺‘(𝐺𝐵)) +s 1s ))
54 oveq1 7438 . . . . . . . . 9 (𝑧 = (𝐺‘(𝐺𝐵)) → (𝑧𝐹𝑤) = ((𝐺‘(𝐺𝐵))𝐹𝑤))
5553, 54opeq12d 4881 . . . . . . . 8 (𝑧 = (𝐺‘(𝐺𝐵)) → ⟨(𝑧 +s 1s ), (𝑧𝐹𝑤)⟩ = ⟨((𝐺‘(𝐺𝐵)) +s 1s ), ((𝐺‘(𝐺𝐵))𝐹𝑤)⟩)
56 oveq2 7439 . . . . . . . . 9 (𝑤 = (2nd ‘(𝑅‘(𝐺𝐵))) → ((𝐺‘(𝐺𝐵))𝐹𝑤) = ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵)))))
5756opeq2d 4880 . . . . . . . 8 (𝑤 = (2nd ‘(𝑅‘(𝐺𝐵))) → ⟨((𝐺‘(𝐺𝐵)) +s 1s ), ((𝐺‘(𝐺𝐵))𝐹𝑤)⟩ = ⟨((𝐺‘(𝐺𝐵)) +s 1s ), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩)
58 oveq1 7438 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥 +s 1s ) = (𝑧 +s 1s ))
59 oveq1 7438 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥𝐹𝑦) = (𝑧𝐹𝑦))
6058, 59opeq12d 4881 . . . . . . . . 9 (𝑥 = 𝑧 → ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩ = ⟨(𝑧 +s 1s ), (𝑧𝐹𝑦)⟩)
61 oveq2 7439 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝑧𝐹𝑦) = (𝑧𝐹𝑤))
6261opeq2d 4880 . . . . . . . . 9 (𝑦 = 𝑤 → ⟨(𝑧 +s 1s ), (𝑧𝐹𝑦)⟩ = ⟨(𝑧 +s 1s ), (𝑧𝐹𝑤)⟩)
6360, 62cbvmpov 7528 . . . . . . . 8 (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩) = (𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 +s 1s ), (𝑧𝐹𝑤)⟩)
64 opex 5469 . . . . . . . 8 ⟨((𝐺‘(𝐺𝐵)) +s 1s ), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩ ∈ V
6555, 57, 63, 64ovmpo 7593 . . . . . . 7 (((𝐺‘(𝐺𝐵)) ∈ V ∧ (2nd ‘(𝑅‘(𝐺𝐵))) ∈ V) → ((𝐺‘(𝐺𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(𝐺𝐵)))) = ⟨((𝐺‘(𝐺𝐵)) +s 1s ), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩)
6651, 52, 65mp2an 692 . . . . . 6 ((𝐺‘(𝐺𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(𝐺𝐵)))) = ⟨((𝐺‘(𝐺𝐵)) +s 1s ), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩
6750, 66eqtrdi 2793 . . . . 5 ((𝜑 ∧ (𝐺𝐵) ∈ ω) → (𝑅‘suc (𝐺𝐵)) = ⟨((𝐺‘(𝐺𝐵)) +s 1s ), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩)
6867fveq2d 6910 . . . 4 ((𝜑 ∧ (𝐺𝐵) ∈ ω) → (2nd ‘(𝑅‘suc (𝐺𝐵))) = (2nd ‘⟨((𝐺‘(𝐺𝐵)) +s 1s ), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩))
69 ovex 7464 . . . . 5 ((𝐺‘(𝐺𝐵)) +s 1s ) ∈ V
70 ovex 7464 . . . . 5 ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵)))) ∈ V
7169, 70op2nd 8023 . . . 4 (2nd ‘⟨((𝐺‘(𝐺𝐵)) +s 1s ), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩) = ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))
7268, 71eqtrdi 2793 . . 3 ((𝜑 ∧ (𝐺𝐵) ∈ ω) → (2nd ‘(𝑅‘suc (𝐺𝐵))) = ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵)))))
7323, 72syldan 591 . 2 ((𝜑𝐵𝑍) → (2nd ‘(𝑅‘suc (𝐺𝐵))) = ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵)))))
741, 2, 3, 4, 5noseqrdglem 28311 . . . . . 6 ((𝜑𝐵𝑍) → ⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩ ∈ ran 𝑅)
7574, 16eleqtrrd 2844 . . . . 5 ((𝜑𝐵𝑍) → ⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩ ∈ 𝑆)
76 funopfv 6958 . . . . 5 (Fun 𝑆 → (⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩ ∈ 𝑆 → (𝑆𝐵) = (2nd ‘(𝑅‘(𝐺𝐵)))))
779, 75, 76sylc 65 . . . 4 ((𝜑𝐵𝑍) → (𝑆𝐵) = (2nd ‘(𝑅‘(𝐺𝐵))))
7877eqcomd 2743 . . 3 ((𝜑𝐵𝑍) → (2nd ‘(𝑅‘(𝐺𝐵))) = (𝑆𝐵))
7930, 78oveq12d 7449 . 2 ((𝜑𝐵𝑍) → ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵)))) = (𝐵𝐹(𝑆𝐵)))
8037, 73, 793eqtrd 2781 1 ((𝜑𝐵𝑍) → (𝑆‘(𝐵 +s 1s )) = (𝐵𝐹(𝑆𝐵)))
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  suc csuc 6386  Fun wfun 6555   Fn wfn 6556  1-1-ontowf1o 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:  seqsp1  28317
  Copyright terms: Public domain W3C validator