Theorem naddcnfid2 43332

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)) ∧ 𝐹 ∈ 𝑆) → ((𝑋 × {∅}) ∘f +o 𝐹) = 𝐹)

Proof of Theorem naddcnfid2

Step	Hyp	Ref	Expression
1		peano1 7929	. . . . . 6 ⊢ ∅ ∈ ω
2		fconst6g 6812	. . . . . 6 ⊢ (∅ ∈ ω → (𝑋 × {∅}):𝑋⟶ω)
3	1, 2	mp1i 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)
5	1	a1i 11	. . . . . 6 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ∅ ∈ ω)
6	4, 5	fczfsuppd 9457	. . . . 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))
8	7	eleq2d 2830	. . . . . 6 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ 𝑆 ↔ (𝑋 × {∅}) ∈ dom (ω CNF 𝑋)))
9		eqid 2740	. . . . . . 7 ⊢ dom (ω CNF 𝑋) = dom (ω CNF 𝑋)
10		omelon 9717	. . . . . . . 8 ⊢ ω ∈ On
11	10	a1i 11	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ω ∈ On)
12	9, 11, 4	cantnfs 9737	. . . . . 6 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ dom (ω CNF 𝑋) ↔ ((𝑋 × {∅}):𝑋⟶ω ∧ (𝑋 × {∅}) finSupp ∅)))
13	8, 12	bitrd 279	. . . . 5 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ 𝑆 ↔ ((𝑋 × {∅}):𝑋⟶ω ∧ (𝑋 × {∅}) finSupp ∅)))
14	3, 6, 13	mpbir2and 712	. . . 4 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}) ∈ 𝑆)
15		naddcnfcom 43330	. . . . 5 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ ((𝑋 × {∅}) ∈ 𝑆 ∧ 𝐹 ∈ 𝑆)) → ((𝑋 × {∅}) ∘f +o 𝐹) = (𝐹 ∘f +o (𝑋 × {∅})))
16	15	ex 412	. . . 4 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (((𝑋 × {∅}) ∈ 𝑆 ∧ 𝐹 ∈ 𝑆) → ((𝑋 × {∅}) ∘f +o 𝐹) = (𝐹 ∘f +o (𝑋 × {∅}))))
17	14, 16	mpand 694	. . 3 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ 𝑆 → ((𝑋 × {∅}) ∘f +o 𝐹) = (𝐹 ∘f +o (𝑋 × {∅}))))
18	17	imp 406	. 2 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) → ((𝑋 × {∅}) ∘f +o 𝐹) = (𝐹 ∘f +o (𝑋 × {∅})))
19		naddcnfid1 43331	. 2 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) → (𝐹 ∘f +o (𝑋 × {∅})) = 𝐹)
20	18, 19	eqtrd 2780	1 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) → ((𝑋 × {∅}) ∘f +o 𝐹) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∅c0 4352  {csn 4648   class class class wbr 5166   × cxp 5698  dom cdm 5700  Oncon0 6397  ⟶wf 6571  (class class class)co 7450   ∘_f cof 7714  ωcom 7905   +_o coa 8521   finSupp cfsupp 9433   CNF ccnf 9732

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 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7772  ax-inf2 9712

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 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6334  df-ord 6400  df-on 6401  df-lim 6402  df-suc 6403  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fo 6581  df-f1o 6582  df-fv 6583  df-ov 7453  df-oprab 7454  df-mpo 7455  df-of 7716  df-om 7906  df-2nd 8033  df-supp 8204  df-frecs 8324  df-wrecs 8355  df-recs 8429  df-rdg 8468  df-seqom 8506  df-oadd 8528  df-map 8888  df-en 9006  df-fin 9009  df-fsupp 9434  df-cnf 9733

This theorem is referenced by: (None)