Theorem naddcnfid1 41284

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 7781	. . . 4 ⊢ ∅ ∈ ω
2		fconst6g 6700	. . . 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 9222	. . 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 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 9481	. . . . . 6 ⊢ ω ∈ On
11	10	a1i 11	. . . . 5 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ω ∈ On)
12	9, 11, 4	cantnfs 9501	. . . 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 710	. 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 9501	. . . . . . . 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 6638	. . . . 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 6638	. . . . 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 772	. . . 4 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) → 𝑋 ∈ On)
24		inidm 4162	. . . 4 ⊢ (𝑋 ∩ 𝑋) = 𝑋
25	20, 22, 23, 23, 24	offn 7587	. . 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 780	. . . . 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 7591	. . . . 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 836	. . . 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 7116	. . . . . 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 7332	. . . 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 7000	. . . . 5 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ ω)
37		nna0 8484	. . . . 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 6951	. 2 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) → (𝐹 ∘f +o (𝑋 × {∅})) = 𝐹)
41	14, 40	mpidan 686	1 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) → (𝐹 ∘f +o (𝑋 × {∅})) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1540   ∈ wcel 2105  ∅c0 4266  {csn 4570   class class class wbr 5086   × cxp 5605  dom cdm 5607  Oncon0 6288   Fn wfn 6460  ⟶wf 6461  ‘cfv 6465  (class class class)co 7316   ∘_f cof 7572  ωcom 7758   +_o coa 8342   finSupp cfsupp 9204   CNF ccnf 9496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5223  ax-sep 5237  ax-nul 5244  ax-pow 5302  ax-pr 5366  ax-un 7629  ax-inf2 9476

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  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 3350  df-rab 3404  df-v 3442  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-pss 3915  df-nul 4267  df-if 4471  df-pw 4546  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4850  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5170  df-tr 5204  df-id 5506  df-eprel 5512  df-po 5520  df-so 5521  df-fr 5562  df-we 5564  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-pred 6224  df-ord 6291  df-on 6292  df-lim 6293  df-suc 6294  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-f1 6470  df-fo 6471  df-f1o 6472  df-fv 6473  df-ov 7319  df-oprab 7320  df-mpo 7321  df-of 7574  df-om 7759  df-2nd 7878  df-supp 8026  df-frecs 8145  df-wrecs 8176  df-recs 8250  df-rdg 8289  df-seqom 8327  df-oadd 8349  df-map 8666  df-en 8783  df-fin 8786  df-fsupp 9205  df-cnf 9497

This theorem is referenced by:  naddcnfid2  41285