Theorem naddcnfid1 41267

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 7767	. . . 4 ⊢ ∅ ∈ ω
2		fconst6g 6693	. . . 4 ⊢ (∅ ∈ ω → (𝑋 × {∅}):𝑋⟶ω)
3	1, 2	mp1i 13	. . 3 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}):𝑋⟶ω)
4		simpl 484	. . . 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 9198	. . 3 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}) finSupp ∅)
7		simpr 486	. . . . 5 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑆 = dom (ω CNF 𝑋))
8	7	eleq2d 2822	. . . 4 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ 𝑆 ↔ (𝑋 × {∅}) ∈ dom (ω CNF 𝑋)))
9		eqid 2736	. . . . 5 ⊢ dom (ω CNF 𝑋) = dom (ω CNF 𝑋)
10		omelon 9456	. . . . . 6 ⊢ ω ∈ On
11	10	a1i 11	. . . . 5 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ω ∈ On)
12	9, 11, 4	cantnfs 9476	. . . 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 711	. 2 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}) ∈ 𝑆)
15	7	eleq2d 2822	. . . . . . . 8 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ 𝑆 ↔ 𝐹 ∈ dom (ω CNF 𝑋)))
16	9, 11, 4	cantnfs 9476	. . . . . . . 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 500	. . . . . 6 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) → 𝐹:𝑋⟶ω)
19	18	ffnd 6631	. . . . 5 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) → 𝐹 Fn 𝑋)
20	19	adantr 482	. . . 4 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) → 𝐹 Fn 𝑋)
21	2	ffnd 6631	. . . . 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 773	. . . 4 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) → 𝑋 ∈ On)
24		inidm 4158	. . . 4 ⊢ (𝑋 ∩ 𝑋) = 𝑋
25	20, 22, 23, 23, 24	offn 7578	. . 3 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) → (𝐹 ∘f +o (𝑋 × {∅})) Fn 𝑋)
26	20	adantr 482	. . . . 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 781	. . . . 5 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥 ∈ 𝑋) → 𝑋 ∈ On)
29		simpr 486	. . . . 5 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
30		fnfvof 7582	. . . . 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 837	. . . 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 7109	. . . . . 6 ⊢ ((∅ ∈ ω ∧ 𝑥 ∈ 𝑋) → ((𝑋 × {∅})‘𝑥) = ∅)
33	1, 29, 32	sylancr 588	. . . . 5 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥 ∈ 𝑋) → ((𝑋 × {∅})‘𝑥) = ∅)
34	33	oveq2d 7323	. . . 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 482	. . . . . 6 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) → 𝐹:𝑋⟶ω)
36	35	ffvelcdmda 6993	. . . . 5 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ ω)
37		nna0 8466	. . . . 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 2780	. . 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 6944	. 2 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) → (𝐹 ∘f +o (𝑋 × {∅})) = 𝐹)
41	14, 40	mpidan 687	1 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) → (𝐹 ∘f +o (𝑋 × {∅})) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1539   ∈ wcel 2104  ∅c0 4262  {csn 4565   class class class wbr 5081   × cxp 5598  dom cdm 5600  Oncon0 6281   Fn wfn 6453  ⟶wf 6454  ‘cfv 6458  (class class class)co 7307   ∘_f cof 7563  ωcom 7744   +_o coa 8325   finSupp cfsupp 9180   CNF ccnf 9471

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620  ax-inf2 9451

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3340  df-rab 3341  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5500  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-we 5557  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-pred 6217  df-ord 6284  df-on 6285  df-lim 6286  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-ov 7310  df-oprab 7311  df-mpo 7312  df-of 7565  df-om 7745  df-2nd 7864  df-supp 8009  df-frecs 8128  df-wrecs 8159  df-recs 8233  df-rdg 8272  df-seqom 8310  df-oadd 8332  df-map 8648  df-en 8765  df-fin 8768  df-fsupp 9181  df-cnf 9472

This theorem is referenced by:  naddcnfid2  41268