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Theorem naddcnfid2 42861
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 7891 . . . . . 6 ∅ ∈ ω
2 fconst6g 6780 . . . . . 6 (∅ ∈ ω → (𝑋 × {∅}):𝑋⟶ω)
31, 2mp1i 13 . . . . 5 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}):𝑋⟶ω)
4 simpl 481 . . . . . 6 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑋 ∈ On)
51a1i 11 . . . . . 6 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ∅ ∈ ω)
64, 5fczfsuppd 9407 . . . . 5 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}) finSupp ∅)
7 simpr 483 . . . . . . 7 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑆 = dom (ω CNF 𝑋))
87eleq2d 2811 . . . . . 6 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ 𝑆 ↔ (𝑋 × {∅}) ∈ dom (ω CNF 𝑋)))
9 eqid 2725 . . . . . . 7 dom (ω CNF 𝑋) = dom (ω CNF 𝑋)
10 omelon 9667 . . . . . . . 8 ω ∈ On
1110a1i 11 . . . . . . 7 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ω ∈ On)
129, 11, 4cantnfs 9687 . . . . . 6 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ dom (ω CNF 𝑋) ↔ ((𝑋 × {∅}):𝑋⟶ω ∧ (𝑋 × {∅}) finSupp ∅)))
138, 12bitrd 278 . . . . 5 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ 𝑆 ↔ ((𝑋 × {∅}):𝑋⟶ω ∧ (𝑋 × {∅}) finSupp ∅)))
143, 6, 13mpbir2and 711 . . . 4 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}) ∈ 𝑆)
15 naddcnfcom 42859 . . . . 5 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ ((𝑋 × {∅}) ∈ 𝑆𝐹𝑆)) → ((𝑋 × {∅}) ∘f +o 𝐹) = (𝐹f +o (𝑋 × {∅})))
1615ex 411 . . . 4 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (((𝑋 × {∅}) ∈ 𝑆𝐹𝑆) → ((𝑋 × {∅}) ∘f +o 𝐹) = (𝐹f +o (𝑋 × {∅}))))
1714, 16mpand 693 . . 3 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹𝑆 → ((𝑋 × {∅}) ∘f +o 𝐹) = (𝐹f +o (𝑋 × {∅}))))
1817imp 405 . 2 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹𝑆) → ((𝑋 × {∅}) ∘f +o 𝐹) = (𝐹f +o (𝑋 × {∅})))
19 naddcnfid1 42860 . 2 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹𝑆) → (𝐹f +o (𝑋 × {∅})) = 𝐹)
2018, 19eqtrd 2765 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 394   = wceq 1533  wcel 2098  c0 4318  {csn 4624   class class class wbr 5143   × cxp 5670  dom cdm 5672  Oncon0 6364  wf 6538  (class class class)co 7415  f cof 7679  ωcom 7867   +o coa 8480   finSupp cfsupp 9383   CNF ccnf 9682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737  ax-inf2 9662
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7418  df-oprab 7419  df-mpo 7420  df-of 7681  df-om 7868  df-2nd 7990  df-supp 8162  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-seqom 8465  df-oadd 8487  df-map 8843  df-en 8961  df-fin 8964  df-fsupp 9384  df-cnf 9683
This theorem is referenced by: (None)
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