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Theorem naddcnfid1 42420
Description: Identity law for component-wise ordinal addition of Cantor normal forms. (Contributed by RP, 3-Jan-2025.)
Assertion
Ref Expression
naddcnfid1 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹𝑆) → (𝐹f +o (𝑋 × {∅})) = 𝐹)

Proof of Theorem naddcnfid1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 peano1 7882 . . . 4 ∅ ∈ ω
2 fconst6g 6781 . . . 4 (∅ ∈ ω → (𝑋 × {∅}):𝑋⟶ω)
31, 2mp1i 13 . . 3 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}):𝑋⟶ω)
4 simpl 482 . . . 4 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑋 ∈ On)
51a1i 11 . . . 4 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ∅ ∈ ω)
64, 5fczfsuppd 9384 . . 3 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}) finSupp ∅)
7 simpr 484 . . . . 5 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑆 = dom (ω CNF 𝑋))
87eleq2d 2818 . . . 4 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ 𝑆 ↔ (𝑋 × {∅}) ∈ dom (ω CNF 𝑋)))
9 eqid 2731 . . . . 5 dom (ω CNF 𝑋) = dom (ω CNF 𝑋)
10 omelon 9644 . . . . . 6 ω ∈ On
1110a1i 11 . . . . 5 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ω ∈ On)
129, 11, 4cantnfs 9664 . . . 4 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ dom (ω CNF 𝑋) ↔ ((𝑋 × {∅}):𝑋⟶ω ∧ (𝑋 × {∅}) finSupp ∅)))
138, 12bitrd 278 . . 3 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ 𝑆 ↔ ((𝑋 × {∅}):𝑋⟶ω ∧ (𝑋 × {∅}) finSupp ∅)))
143, 6, 13mpbir2and 710 . 2 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}) ∈ 𝑆)
157eleq2d 2818 . . . . . . . 8 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹𝑆𝐹 ∈ dom (ω CNF 𝑋)))
169, 11, 4cantnfs 9664 . . . . . . . 8 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ dom (ω CNF 𝑋) ↔ (𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅)))
1715, 16bitrd 278 . . . . . . 7 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹𝑆 ↔ (𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅)))
1817simprbda 498 . . . . . 6 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹𝑆) → 𝐹:𝑋⟶ω)
1918ffnd 6719 . . . . 5 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹𝑆) → 𝐹 Fn 𝑋)
2019adantr 480 . . . 4 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) → 𝐹 Fn 𝑋)
212ffnd 6719 . . . . 5 (∅ ∈ ω → (𝑋 × {∅}) Fn 𝑋)
221, 21mp1i 13 . . . 4 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) → (𝑋 × {∅}) Fn 𝑋)
23 simplll 772 . . . 4 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) → 𝑋 ∈ On)
24 inidm 4219 . . . 4 (𝑋𝑋) = 𝑋
2520, 22, 23, 23, 24offn 7686 . . 3 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) → (𝐹f +o (𝑋 × {∅})) Fn 𝑋)
2620adantr 480 . . . . 5 (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥𝑋) → 𝐹 Fn 𝑋)
271, 21mp1i 13 . . . . 5 (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥𝑋) → (𝑋 × {∅}) Fn 𝑋)
28 simp-4l 780 . . . . 5 (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥𝑋) → 𝑋 ∈ On)
29 simpr 484 . . . . 5 (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥𝑋) → 𝑥𝑋)
30 fnfvof 7690 . . . . 5 (((𝐹 Fn 𝑋 ∧ (𝑋 × {∅}) Fn 𝑋) ∧ (𝑋 ∈ On ∧ 𝑥𝑋)) → ((𝐹f +o (𝑋 × {∅}))‘𝑥) = ((𝐹𝑥) +o ((𝑋 × {∅})‘𝑥)))
3126, 27, 28, 29, 30syl22anc 836 . . . 4 (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥𝑋) → ((𝐹f +o (𝑋 × {∅}))‘𝑥) = ((𝐹𝑥) +o ((𝑋 × {∅})‘𝑥)))
32 fvconst2g 7206 . . . . . 6 ((∅ ∈ ω ∧ 𝑥𝑋) → ((𝑋 × {∅})‘𝑥) = ∅)
331, 29, 32sylancr 586 . . . . 5 (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥𝑋) → ((𝑋 × {∅})‘𝑥) = ∅)
3433oveq2d 7428 . . . 4 (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥𝑋) → ((𝐹𝑥) +o ((𝑋 × {∅})‘𝑥)) = ((𝐹𝑥) +o ∅))
3518adantr 480 . . . . . 6 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) → 𝐹:𝑋⟶ω)
3635ffvelcdmda 7087 . . . . 5 (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥𝑋) → (𝐹𝑥) ∈ ω)
37 nna0 8607 . . . . 5 ((𝐹𝑥) ∈ ω → ((𝐹𝑥) +o ∅) = (𝐹𝑥))
3836, 37syl 17 . . . 4 (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥𝑋) → ((𝐹𝑥) +o ∅) = (𝐹𝑥))
3931, 34, 383eqtrd 2775 . . 3 (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥𝑋) → ((𝐹f +o (𝑋 × {∅}))‘𝑥) = (𝐹𝑥))
4025, 20, 39eqfnfvd 7036 . 2 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) → (𝐹f +o (𝑋 × {∅})) = 𝐹)
4114, 40mpidan 686 1 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹𝑆) → (𝐹f +o (𝑋 × {∅})) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  c0 4323  {csn 4629   class class class wbr 5149   × cxp 5675  dom cdm 5677  Oncon0 6365   Fn wfn 6539  wf 6540  cfv 6544  (class class class)co 7412  f cof 7671  ωcom 7858   +o coa 8466   finSupp cfsupp 9364   CNF ccnf 9659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-inf2 9639
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7673  df-om 7859  df-2nd 7979  df-supp 8150  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-seqom 8451  df-oadd 8473  df-map 8825  df-en 8943  df-fin 8946  df-fsupp 9365  df-cnf 9660
This theorem is referenced by:  naddcnfid2  42421
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