Theorem naddcnfid2 44562

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 7924	. . . . . 6 ⊢ ∅ ∈ ω
2		fconst6g 6793	. . . . . 6 ⊢ (∅ ∈ ω → (𝑋 × {∅}):𝑋⟶ω)
3	1, 2	mp1i 14	. . . . 5 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}):𝑋⟶ω)
4		simpl 488	. . . . . 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 9405	. . . . 5 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}) finSupp ∅)
7		simpr 490	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑆 = dom (ω CNF 𝑋))
8	7	eleq2d 2850	. . . . . 6 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ 𝑆 ↔ (𝑋 × {∅}) ∈ dom (ω CNF 𝑋)))
9		eqid 2763	. . . . . . 7 ⊢ dom (ω CNF 𝑋) = dom (ω CNF 𝑋)
10		omelon 9674	. . . . . . . 8 ⊢ ω ∈ On
11	10	a1i 11	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ω ∈ On)
12	9, 11, 4	cantnfs 9694	. . . . . 6 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ dom (ω CNF 𝑋) ↔ ((𝑋 × {∅}):𝑋⟶ω ∧ (𝑋 × {∅}) finSupp ∅)))
13	8, 12	bitrd 282	. . . . 5 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ 𝑆 ↔ ((𝑋 × {∅}):𝑋⟶ω ∧ (𝑋 × {∅}) finSupp ∅)))
14	3, 6, 13	mpbir2and 726	. . . 4 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}) ∈ 𝑆)
15		naddcnfcom 44560	. . . . 5 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ ((𝑋 × {∅}) ∈ 𝑆 ∧ 𝐹 ∈ 𝑆)) → ((𝑋 × {∅}) ∘f +o 𝐹) = (𝐹 ∘f +o (𝑋 × {∅})))
16	15	ex 418	. . . 4 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (((𝑋 × {∅}) ∈ 𝑆 ∧ 𝐹 ∈ 𝑆) → ((𝑋 × {∅}) ∘f +o 𝐹) = (𝐹 ∘f +o (𝑋 × {∅}))))
17	14, 16	mpand 708	. . 3 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ 𝑆 → ((𝑋 × {∅}) ∘f +o 𝐹) = (𝐹 ∘f +o (𝑋 × {∅}))))
18	17	imp 412	. 2 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) → ((𝑋 × {∅}) ∘f +o 𝐹) = (𝐹 ∘f +o (𝑋 × {∅})))
19		naddcnfid1 44561	. 2 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) → (𝐹 ∘f +o (𝑋 × {∅})) = 𝐹)
20	18, 19	eqtrd 2798	1 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) → ((𝑋 × {∅}) ∘f +o 𝐹) = 𝐹)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∅c0 4281  {csn 4588   class class class wbr 5106   × cxp 5653  dom cdm 5655  Oncon0 6380  ⟶wf 6554  (class class class)co 7443   ∘_f cof 7715  ωcom 7901   +_o coa 8501   finSupp cfsupp 9381   CNF ccnf 9689

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-inf2 9669

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-of 7717  df-om 7902  df-2nd 8026  df-supp 8205  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-seqom 8486  df-oadd 8508  df-map 8877  df-en 9002  df-fin 9005  df-fsupp 9382  df-cnf 9690

This theorem is used by: (None)