Theorem naddcnfid1 42199

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 7881	. . . 4 ⊢ ∅ ∈ ω
2		fconst6g 6780	. . . 4 ⊢ (∅ ∈ ω → (𝑋 × {∅}):𝑋⟶ω)
3	1, 2	mp1i 13	. . 3 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}):𝑋⟶ω)
4		simpl 483	. . . 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 9383	. . 3 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}) finSupp ∅)
7		simpr 485	. . . . 5 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑆 = dom (ω CNF 𝑋))
8	7	eleq2d 2819	. . . 4 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ 𝑆 ↔ (𝑋 × {∅}) ∈ dom (ω CNF 𝑋)))
9		eqid 2732	. . . . 5 ⊢ dom (ω CNF 𝑋) = dom (ω CNF 𝑋)
10		omelon 9643	. . . . . 6 ⊢ ω ∈ On
11	10	a1i 11	. . . . 5 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ω ∈ On)
12	9, 11, 4	cantnfs 9663	. . . 4 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ dom (ω CNF 𝑋) ↔ ((𝑋 × {∅}):𝑋⟶ω ∧ (𝑋 × {∅}) finSupp ∅)))
13	8, 12	bitrd 278	. . 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 2819	. . . . . . . 8 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ 𝑆 ↔ 𝐹 ∈ dom (ω CNF 𝑋)))
16	9, 11, 4	cantnfs 9663	. . . . . . . 8 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ dom (ω CNF 𝑋) ↔ (𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅)))
17	15, 16	bitrd 278	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ 𝑆 ↔ (𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅)))
18	17	simprbda 499	. . . . . 6 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) → 𝐹:𝑋⟶ω)
19	18	ffnd 6718	. . . . 5 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) → 𝐹 Fn 𝑋)
20	19	adantr 481	. . . 4 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) → 𝐹 Fn 𝑋)
21	2	ffnd 6718	. . . . 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 4218	. . . 4 ⊢ (𝑋 ∩ 𝑋) = 𝑋
25	20, 22, 23, 23, 24	offn 7685	. . 3 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) → (𝐹 ∘f +o (𝑋 × {∅})) Fn 𝑋)
26	20	adantr 481	. . . . 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 485	. . . . 5 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
30		fnfvof 7689	. . . . 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 7205	. . . . . 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 7427	. . . 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 481	. . . . . 6 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) → 𝐹:𝑋⟶ω)
36	35	ffvelcdmda 7086	. . . . 5 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ ω)
37		nna0 8606	. . . . 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 2776	. . 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 7035	. 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 396   = wceq 1541   ∈ wcel 2106  ∅c0 4322  {csn 4628   class class class wbr 5148   × cxp 5674  dom cdm 5676  Oncon0 6364   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  (class class class)co 7411   ∘_f cof 7670  ωcom 7857   +_o coa 8465   finSupp cfsupp 9363   CNF ccnf 9658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-inf2 9638

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-om 7858  df-2nd 7978  df-supp 8149  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-seqom 8450  df-oadd 8472  df-map 8824  df-en 8942  df-fin 8945  df-fsupp 9364  df-cnf 9659

This theorem is referenced by:  naddcnfid2  42200