Theorem naddcnfid1 43340

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

Proof of Theorem naddcnfid1

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano1 7829	. . . 4 ⊢ ∅ ∈ ω
2		fconst6g 6717	. . . 4 ⊢ (∅ ∈ ω → (𝑋 × {∅}):𝑋⟶ω)
3	1, 2	mp1i 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)
5	1	a1i 11	. . . 4 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ∅ ∈ ω)
6	4, 5	fczfsuppd 9295	. . 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))
8	7	eleq2d 2814	. . . 4 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ 𝑆 ↔ (𝑋 × {∅}) ∈ dom (ω CNF 𝑋)))
9		eqid 2729	. . . . 5 ⊢ dom (ω CNF 𝑋) = dom (ω CNF 𝑋)
10		omelon 9561	. . . . . 6 ⊢ ω ∈ On
11	10	a1i 11	. . . . 5 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ω ∈ On)
12	9, 11, 4	cantnfs 9581	. . . 4 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ dom (ω CNF 𝑋) ↔ ((𝑋 × {∅}):𝑋⟶ω ∧ (𝑋 × {∅}) finSupp ∅)))
13	8, 12	bitrd 279	. . 3 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ 𝑆 ↔ ((𝑋 × {∅}):𝑋⟶ω ∧ (𝑋 × {∅}) finSupp ∅)))
14	3, 6, 13	mpbir2and 713	. 2 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}) ∈ 𝑆)
15	7	eleq2d 2814	. . . . . . . 8 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ 𝑆 ↔ 𝐹 ∈ dom (ω CNF 𝑋)))
16	9, 11, 4	cantnfs 9581	. . . . . . . 8 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ dom (ω CNF 𝑋) ↔ (𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅)))
17	15, 16	bitrd 279	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ 𝑆 ↔ (𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅)))
18	17	simprbda 498	. . . . . 6 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) → 𝐹:𝑋⟶ω)
19	18	ffnd 6657	. . . . 5 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) → 𝐹 Fn 𝑋)
20	19	adantr 480	. . . 4 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) → 𝐹 Fn 𝑋)
21	2	ffnd 6657	. . . . 5 ⊢ (∅ ∈ ω → (𝑋 × {∅}) Fn 𝑋)
22	1, 21	mp1i 13	. . . 4 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) → (𝑋 × {∅}) Fn 𝑋)
23		simplll 774	. . . 4 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) → 𝑋 ∈ On)
24		inidm 4180	. . . 4 ⊢ (𝑋 ∩ 𝑋) = 𝑋
25	20, 22, 23, 23, 24	offn 7630	. . 3 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) → (𝐹 ∘f +o (𝑋 × {∅})) Fn 𝑋)
26	20	adantr 480	. . . . 5 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥 ∈ 𝑋) → 𝐹 Fn 𝑋)
27	1, 21	mp1i 13	. . . . 5 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥 ∈ 𝑋) → (𝑋 × {∅}) Fn 𝑋)
28		simp-4l 782	. . . . 5 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥 ∈ 𝑋) → 𝑋 ∈ On)
29		simpr 484	. . . . 5 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
30		fnfvof 7634	. . . . 5 ⊢ (((𝐹 Fn 𝑋 ∧ (𝑋 × {∅}) Fn 𝑋) ∧ (𝑋 ∈ On ∧ 𝑥 ∈ 𝑋)) → ((𝐹 ∘f +o (𝑋 × {∅}))‘𝑥) = ((𝐹‘𝑥) +o ((𝑋 × {∅})‘𝑥)))
31	26, 27, 28, 29, 30	syl22anc 838	. . . 4 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥 ∈ 𝑋) → ((𝐹 ∘f +o (𝑋 × {∅}))‘𝑥) = ((𝐹‘𝑥) +o ((𝑋 × {∅})‘𝑥)))
32		fvconst2g 7142	. . . . . 6 ⊢ ((∅ ∈ ω ∧ 𝑥 ∈ 𝑋) → ((𝑋 × {∅})‘𝑥) = ∅)
33	1, 29, 32	sylancr 587	. . . . 5 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥 ∈ 𝑋) → ((𝑋 × {∅})‘𝑥) = ∅)
34	33	oveq2d 7369	. . . 4 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) +o ((𝑋 × {∅})‘𝑥)) = ((𝐹‘𝑥) +o ∅))
35	18	adantr 480	. . . . . 6 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) → 𝐹:𝑋⟶ω)
36	35	ffvelcdmda 7022	. . . . 5 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ ω)
37		nna0 8529	. . . . 5 ⊢ ((𝐹‘𝑥) ∈ ω → ((𝐹‘𝑥) +o ∅) = (𝐹‘𝑥))
38	36, 37	syl 17	. . . 4 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) +o ∅) = (𝐹‘𝑥))
39	31, 34, 38	3eqtrd 2768	. . 3 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥 ∈ 𝑋) → ((𝐹 ∘f +o (𝑋 × {∅}))‘𝑥) = (𝐹‘𝑥))
40	25, 20, 39	eqfnfvd 6972	. 2 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) → (𝐹 ∘f +o (𝑋 × {∅})) = 𝐹)
41	14, 40	mpidan 689	1 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) → (𝐹 ∘f +o (𝑋 × {∅})) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∅c0 4286  {csn 4579   class class class wbr 5095   × cxp 5621  dom cdm 5623  Oncon0 6311   Fn wfn 6481  ⟶wf 6482  ‘cfv 6486  (class class class)co 7353   ∘_f cof 7615  ωcom 7806   +_o coa 8392   finSupp cfsupp 9270   CNF ccnf 9576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-inf2 9556

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-of 7617  df-om 7807  df-2nd 7932  df-supp 8101  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-seqom 8377  df-oadd 8399  df-map 8762  df-en 8880  df-fin 8883  df-fsupp 9271  df-cnf 9577

This theorem is referenced by:  naddcnfid2  43341