Theorem tfsconcat0b 43364

Description: The concatentation with the empty series leaves the finite series unchanged. (Contributed by RP, 1-Mar-2025.)

Hypothesis

Ref	Expression
tfsconcat.op	⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))}))

Assertion

Ref	Expression
tfsconcat0b	⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → (𝐴 = ∅ ↔ (𝐴 + 𝐵) = 𝐵))

Distinct variable groups:   𝐴,𝑎,𝑏,𝑥,𝑦,𝑧   𝐵,𝑎,𝑏,𝑥,𝑦,𝑧   𝐶,𝑎,𝑏,𝑥,𝑦,𝑧   𝐷,𝑎,𝑏,𝑥,𝑦,𝑧

Allowed substitution hints:   + (𝑥,𝑦,𝑧,𝑎,𝑏)

Proof of Theorem tfsconcat0b

Step	Hyp	Ref	Expression
1		nnon 7894	. . . . 5 ⊢ (𝐷 ∈ ω → 𝐷 ∈ On)
2	1	anim2i 617	. . . 4 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (𝐶 ∈ On ∧ 𝐷 ∈ On))
3	2	anim2i 617	. . 3 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → ((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)))
4		tfsconcat.op	. . . 4 ⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))}))
5	4	tfsconcat0i 43363	. . 3 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 = ∅ → (𝐴 + 𝐵) = 𝐵))
6	3, 5	syl 17	. 2 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → (𝐴 = ∅ → (𝐴 + 𝐵) = 𝐵))
7		dmeq 5913	. . 3 ⊢ ((𝐴 + 𝐵) = 𝐵 → dom (𝐴 + 𝐵) = dom 𝐵)
8		nna0r 8648	. . . . . . . . 9 ⊢ (𝐷 ∈ ω → (∅ +o 𝐷) = 𝐷)
9	8	adantl 481	. . . . . . . 8 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (∅ +o 𝐷) = 𝐷)
10	9	eqeq2d 2747	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → ((𝐶 +o 𝐷) = (∅ +o 𝐷) ↔ (𝐶 +o 𝐷) = 𝐷))
11		eqcom 2743	. . . . . . 7 ⊢ ((𝐶 +o 𝐷) = (∅ +o 𝐷) ↔ (∅ +o 𝐷) = (𝐶 +o 𝐷))
12	10, 11	bitr3di 286	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → ((𝐶 +o 𝐷) = 𝐷 ↔ (∅ +o 𝐷) = (𝐶 +o 𝐷)))
13		on0eln0 6439	. . . . . . . . . 10 ⊢ (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
14	13	adantr 480	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
15		df-ne 2940	. . . . . . . . 9 ⊢ (𝐶 ≠ ∅ ↔ ¬ 𝐶 = ∅)
16	14, 15	bitr2di 288	. . . . . . . 8 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (¬ 𝐶 = ∅ ↔ ∅ ∈ 𝐶))
17		peano1 7911	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ ω
18		nnaordr 8659	. . . . . . . . . . . . . . 15 ⊢ ((∅ ∈ ω ∧ 𝐶 ∈ ω ∧ 𝐷 ∈ ω) → (∅ ∈ 𝐶 ↔ (∅ +o 𝐷) ∈ (𝐶 +o 𝐷)))
19	17, 18	mp3an1 1449	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ ω ∧ 𝐷 ∈ ω) → (∅ ∈ 𝐶 ↔ (∅ +o 𝐷) ∈ (𝐶 +o 𝐷)))
20	19	biimpd 229	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ω ∧ 𝐷 ∈ ω) → (∅ ∈ 𝐶 → (∅ +o 𝐷) ∈ (𝐶 +o 𝐷)))
21	20	ex 412	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ ω → (𝐷 ∈ ω → (∅ ∈ 𝐶 → (∅ +o 𝐷) ∈ (𝐶 +o 𝐷))))
22	21	a1i 11	. . . . . . . . . . 11 ⊢ (𝐶 ∈ On → (𝐶 ∈ ω → (𝐷 ∈ ω → (∅ ∈ 𝐶 → (∅ +o 𝐷) ∈ (𝐶 +o 𝐷)))))
23		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ ω) ∧ ω ⊆ 𝐶) → ω ⊆ 𝐶)
24		oaword1 8591	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → 𝐶 ⊆ (𝐶 +o 𝐷))
25	2, 24	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → 𝐶 ⊆ (𝐶 +o 𝐷))
26	25	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ ω) ∧ ω ⊆ 𝐶) → 𝐶 ⊆ (𝐶 +o 𝐷))
27	23, 26	sstrd 3993	. . . . . . . . . . . . . . 15 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ ω) ∧ ω ⊆ 𝐶) → ω ⊆ (𝐶 +o 𝐷))
28		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝐷 ∈ ω → 𝐷 ∈ ω)
29	8, 28	eqeltrd 2840	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 ∈ ω → (∅ +o 𝐷) ∈ ω)
30	29	ad2antlr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ ω) ∧ ω ⊆ 𝐶) → (∅ +o 𝐷) ∈ ω)
31	27, 30	sseldd 3983	. . . . . . . . . . . . . 14 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ ω) ∧ ω ⊆ 𝐶) → (∅ +o 𝐷) ∈ (𝐶 +o 𝐷))
32	31	a1d 25	. . . . . . . . . . . . 13 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ ω) ∧ ω ⊆ 𝐶) → (∅ ∈ 𝐶 → (∅ +o 𝐷) ∈ (𝐶 +o 𝐷)))
33	32	exp31 419	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ On → (𝐷 ∈ ω → (ω ⊆ 𝐶 → (∅ ∈ 𝐶 → (∅ +o 𝐷) ∈ (𝐶 +o 𝐷)))))
34	33	com23 86	. . . . . . . . . . 11 ⊢ (𝐶 ∈ On → (ω ⊆ 𝐶 → (𝐷 ∈ ω → (∅ ∈ 𝐶 → (∅ +o 𝐷) ∈ (𝐶 +o 𝐷)))))
35		eloni 6393	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ On → Ord 𝐶)
36		ordom 7898	. . . . . . . . . . . 12 ⊢ Ord ω
37		ordtri2or 6481	. . . . . . . . . . . 12 ⊢ ((Ord 𝐶 ∧ Ord ω) → (𝐶 ∈ ω ∨ ω ⊆ 𝐶))
38	35, 36, 37	sylancl 586	. . . . . . . . . . 11 ⊢ (𝐶 ∈ On → (𝐶 ∈ ω ∨ ω ⊆ 𝐶))
39	22, 34, 38	mpjaod 860	. . . . . . . . . 10 ⊢ (𝐶 ∈ On → (𝐷 ∈ ω → (∅ ∈ 𝐶 → (∅ +o 𝐷) ∈ (𝐶 +o 𝐷))))
40	39	imp 406	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (∅ ∈ 𝐶 → (∅ +o 𝐷) ∈ (𝐶 +o 𝐷)))
41		elneq 9639	. . . . . . . . . 10 ⊢ ((∅ +o 𝐷) ∈ (𝐶 +o 𝐷) → (∅ +o 𝐷) ≠ (𝐶 +o 𝐷))
42	41	neneqd 2944	. . . . . . . . 9 ⊢ ((∅ +o 𝐷) ∈ (𝐶 +o 𝐷) → ¬ (∅ +o 𝐷) = (𝐶 +o 𝐷))
43	40, 42	syl6 35	. . . . . . . 8 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (∅ ∈ 𝐶 → ¬ (∅ +o 𝐷) = (𝐶 +o 𝐷)))
44	16, 43	sylbid 240	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (¬ 𝐶 = ∅ → ¬ (∅ +o 𝐷) = (𝐶 +o 𝐷)))
45	44	con4d 115	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → ((∅ +o 𝐷) = (𝐶 +o 𝐷) → 𝐶 = ∅))
46	12, 45	sylbid 240	. . . . 5 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → ((𝐶 +o 𝐷) = 𝐷 → 𝐶 = ∅))
47	46	adantl 481	. . . 4 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → ((𝐶 +o 𝐷) = 𝐷 → 𝐶 = ∅))
48	4	tfsconcatfn 43356	. . . . . . 7 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 + 𝐵) Fn (𝐶 +o 𝐷))
49	3, 48	syl 17	. . . . . 6 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → (𝐴 + 𝐵) Fn (𝐶 +o 𝐷))
50	49	fndmd 6672	. . . . 5 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → dom (𝐴 + 𝐵) = (𝐶 +o 𝐷))
51		fndm 6670	. . . . . 6 ⊢ (𝐵 Fn 𝐷 → dom 𝐵 = 𝐷)
52	51	ad2antlr 727	. . . . 5 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → dom 𝐵 = 𝐷)
53	50, 52	eqeq12d 2752	. . . 4 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → (dom (𝐴 + 𝐵) = dom 𝐵 ↔ (𝐶 +o 𝐷) = 𝐷))
54		fnrel 6669	. . . . . . . 8 ⊢ (𝐴 Fn 𝐶 → Rel 𝐴)
55		reldm0 5937	. . . . . . . 8 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅))
56	54, 55	syl 17	. . . . . . 7 ⊢ (𝐴 Fn 𝐶 → (𝐴 = ∅ ↔ dom 𝐴 = ∅))
57		fndm 6670	. . . . . . . 8 ⊢ (𝐴 Fn 𝐶 → dom 𝐴 = 𝐶)
58	57	eqeq1d 2738	. . . . . . 7 ⊢ (𝐴 Fn 𝐶 → (dom 𝐴 = ∅ ↔ 𝐶 = ∅))
59	56, 58	bitrd 279	. . . . . 6 ⊢ (𝐴 Fn 𝐶 → (𝐴 = ∅ ↔ 𝐶 = ∅))
60	59	adantr 480	. . . . 5 ⊢ ((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) → (𝐴 = ∅ ↔ 𝐶 = ∅))
61	60	adantr 480	. . . 4 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → (𝐴 = ∅ ↔ 𝐶 = ∅))
62	47, 53, 61	3imtr4d 294	. . 3 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → (dom (𝐴 + 𝐵) = dom 𝐵 → 𝐴 = ∅))
63	7, 62	syl5 34	. 2 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → ((𝐴 + 𝐵) = 𝐵 → 𝐴 = ∅))
64	6, 63	impbid 212	1 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → (𝐴 = ∅ ↔ (𝐴 + 𝐵) = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1539   ∈ wcel 2107   ≠ wne 2939  ∃wrex 3069  Vcvv 3479   ∖ cdif 3947   ∪ cun 3948   ⊆ wss 3950  ∅c0 4332  {copab 5204  dom cdm 5684  Rel wrel 5689  Ord word 6382  Oncon0 6383   Fn wfn 6555  ‘cfv 6560  (class class class)co 7432   ∈ cmpo 7434  ωcom 7888   +_o coa 8504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-reg 9633

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-oadd 8511

This theorem is referenced by: (None)