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Theorem naddcnfid2 43270
Description: Identity law for component-wise ordinal addition of Cantor normal forms. (Contributed by RP, 3-Jan-2025.)
Assertion
Ref Expression
naddcnfid2 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹𝑆) → ((𝑋 × {∅}) ∘f +o 𝐹) = 𝐹)

Proof of Theorem naddcnfid2
StepHypRef Expression
1 peano1 7923 . . . . . 6 ∅ ∈ ω
2 fconst6g 6809 . . . . . 6 (∅ ∈ ω → (𝑋 × {∅}):𝑋⟶ω)
31, 2mp1i 13 . . . . 5 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}):𝑋⟶ω)
4 simpl 482 . . . . . 6 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑋 ∈ On)
51a1i 11 . . . . . 6 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ∅ ∈ ω)
64, 5fczfsuppd 9451 . . . . 5 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}) finSupp ∅)
7 simpr 484 . . . . . . 7 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑆 = dom (ω CNF 𝑋))
87eleq2d 2824 . . . . . 6 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ 𝑆 ↔ (𝑋 × {∅}) ∈ dom (ω CNF 𝑋)))
9 eqid 2734 . . . . . . 7 dom (ω CNF 𝑋) = dom (ω CNF 𝑋)
10 omelon 9711 . . . . . . . 8 ω ∈ On
1110a1i 11 . . . . . . 7 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ω ∈ On)
129, 11, 4cantnfs 9731 . . . . . 6 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ dom (ω CNF 𝑋) ↔ ((𝑋 × {∅}):𝑋⟶ω ∧ (𝑋 × {∅}) finSupp ∅)))
138, 12bitrd 279 . . . . 5 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ 𝑆 ↔ ((𝑋 × {∅}):𝑋⟶ω ∧ (𝑋 × {∅}) finSupp ∅)))
143, 6, 13mpbir2and 712 . . . 4 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}) ∈ 𝑆)
15 naddcnfcom 43268 . . . . 5 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ ((𝑋 × {∅}) ∈ 𝑆𝐹𝑆)) → ((𝑋 × {∅}) ∘f +o 𝐹) = (𝐹f +o (𝑋 × {∅})))
1615ex 412 . . . 4 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (((𝑋 × {∅}) ∈ 𝑆𝐹𝑆) → ((𝑋 × {∅}) ∘f +o 𝐹) = (𝐹f +o (𝑋 × {∅}))))
1714, 16mpand 694 . . 3 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹𝑆 → ((𝑋 × {∅}) ∘f +o 𝐹) = (𝐹f +o (𝑋 × {∅}))))
1817imp 406 . 2 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹𝑆) → ((𝑋 × {∅}) ∘f +o 𝐹) = (𝐹f +o (𝑋 × {∅})))
19 naddcnfid1 43269 . 2 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹𝑆) → (𝐹f +o (𝑋 × {∅})) = 𝐹)
2018, 19eqtrd 2774 1 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹𝑆) → ((𝑋 × {∅}) ∘f +o 𝐹) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2103  c0 4347  {csn 4648   class class class wbr 5169   × cxp 5697  dom cdm 5699  Oncon0 6394  wf 6568  (class class class)co 7445  f cof 7708  ωcom 7899   +o coa 8515   finSupp cfsupp 9427   CNF ccnf 9726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766  ax-inf2 9706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-tr 5287  df-id 5597  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-we 5656  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-pred 6331  df-ord 6397  df-on 6398  df-lim 6399  df-suc 6400  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-ov 7448  df-oprab 7449  df-mpo 7450  df-of 7710  df-om 7900  df-2nd 8027  df-supp 8198  df-frecs 8318  df-wrecs 8349  df-recs 8423  df-rdg 8462  df-seqom 8500  df-oadd 8522  df-map 8882  df-en 9000  df-fin 9003  df-fsupp 9428  df-cnf 9727
This theorem is referenced by: (None)
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