Theorem tfsconcat0b 43861

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 7837	. . . . 5 ⊢ (𝐷 ∈ ω → 𝐷 ∈ On)
2	1	anim2i 625	. . . 4 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (𝐶 ∈ On ∧ 𝐷 ∈ On))
3	2	anim2i 625	. . 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 43860	. . 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 5868	. . 3 ⊢ ((𝐴 + 𝐵) = 𝐵 → dom (𝐴 + 𝐵) = dom 𝐵)
8		nna0r 8563	. . . . . . . . 9 ⊢ (𝐷 ∈ ω → (∅ +o 𝐷) = 𝐷)
9	8	adantl 484	. . . . . . . 8 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (∅ +o 𝐷) = 𝐷)
10	9	eqeq2d 2763	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → ((𝐶 +o 𝐷) = (∅ +o 𝐷) ↔ (𝐶 +o 𝐷) = 𝐷))
11		eqcom 2759	. . . . . . 7 ⊢ ((𝐶 +o 𝐷) = (∅ +o 𝐷) ↔ (∅ +o 𝐷) = (𝐶 +o 𝐷))
12	10, 11	bitr3di 288	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → ((𝐶 +o 𝐷) = 𝐷 ↔ (∅ +o 𝐷) = (𝐶 +o 𝐷)))
13		on0eln0 6388	. . . . . . . . . 10 ⊢ (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
14	13	adantr 483	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
15		df-ne 2948	. . . . . . . . 9 ⊢ (𝐶 ≠ ∅ ↔ ¬ 𝐶 = ∅)
16	14, 15	bitr2di 290	. . . . . . . 8 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (¬ 𝐶 = ∅ ↔ ∅ ∈ 𝐶))
17		peano1 7854	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ ω
18		nnaordr 8574	. . . . . . . . . . . . . . 15 ⊢ ((∅ ∈ ω ∧ 𝐶 ∈ ω ∧ 𝐷 ∈ ω) → (∅ ∈ 𝐶 ↔ (∅ +o 𝐷) ∈ (𝐶 +o 𝐷)))
19	17, 18	mp3an1 1459	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ ω ∧ 𝐷 ∈ ω) → (∅ ∈ 𝐶 ↔ (∅ +o 𝐷) ∈ (𝐶 +o 𝐷)))
20	19	biimpd 231	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ω ∧ 𝐷 ∈ ω) → (∅ ∈ 𝐶 → (∅ +o 𝐷) ∈ (𝐶 +o 𝐷)))
21	20	ex 415	. . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . 16 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ ω) ∧ ω ⊆ 𝐶) → ω ⊆ 𝐶)
24		oaword1 8505	. . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . 16 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ ω) ∧ ω ⊆ 𝐶) → 𝐶 ⊆ (𝐶 +o 𝐷))
27	23, 26	sstrd 3937	. . . . . . . . . . . . . . 15 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ ω) ∧ ω ⊆ 𝐶) → ω ⊆ (𝐶 +o 𝐷))
28		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝐷 ∈ ω → 𝐷 ∈ ω)
29	8, 28	eqeltrd 2852	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 ∈ ω → (∅ +o 𝐷) ∈ ω)
30	29	ad2antlr 735	. . . . . . . . . . . . . . 15 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ ω) ∧ ω ⊆ 𝐶) → (∅ +o 𝐷) ∈ ω)
31	27, 30	sseldd 3928	. . . . . . . . . . . . . 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 422	. . . . . . . . . . . 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 6341	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ On → Ord 𝐶)
36		ordom 7841	. . . . . . . . . . . 12 ⊢ Ord ω
37		ordtri2or 6431	. . . . . . . . . . . 12 ⊢ ((Ord 𝐶 ∧ Ord ω) → (𝐶 ∈ ω ∨ ω ⊆ 𝐶))
38	35, 36, 37	sylancl 594	. . . . . . . . . . 11 ⊢ (𝐶 ∈ On → (𝐶 ∈ ω ∨ ω ⊆ 𝐶))
39	22, 34, 38	mpjaod 869	. . . . . . . . . 10 ⊢ (𝐶 ∈ On → (𝐷 ∈ ω → (∅ ∈ 𝐶 → (∅ +o 𝐷) ∈ (𝐶 +o 𝐷))))
40	39	imp 409	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (∅ ∈ 𝐶 → (∅ +o 𝐷) ∈ (𝐶 +o 𝐷)))
41		elneq 9535	. . . . . . . . . 10 ⊢ ((∅ +o 𝐷) ∈ (𝐶 +o 𝐷) → (∅ +o 𝐷) ≠ (𝐶 +o 𝐷))
42	41	neneqd 2952	. . . . . . . . 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 242	. . . . . . 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 242	. . . . 5 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → ((𝐶 +o 𝐷) = 𝐷 → 𝐶 = ∅))
47	46	adantl 484	. . . 4 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → ((𝐶 +o 𝐷) = 𝐷 → 𝐶 = ∅))
48	4	tfsconcatfn 43853	. . . . . . 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 6611	. . . . 5 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → dom (𝐴 + 𝐵) = (𝐶 +o 𝐷))
51		fndm 6609	. . . . . 6 ⊢ (𝐵 Fn 𝐷 → dom 𝐵 = 𝐷)
52	51	ad2antlr 735	. . . . 5 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → dom 𝐵 = 𝐷)
53	50, 52	eqeq12d 2768	. . . 4 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → (dom (𝐴 + 𝐵) = dom 𝐵 ↔ (𝐶 +o 𝐷) = 𝐷))
54		fnrel 6608	. . . . . . . 8 ⊢ (𝐴 Fn 𝐶 → Rel 𝐴)
55		reldm0 5893	. . . . . . . 8 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅))
56	54, 55	syl 17	. . . . . . 7 ⊢ (𝐴 Fn 𝐶 → (𝐴 = ∅ ↔ dom 𝐴 = ∅))
57		fndm 6609	. . . . . . . 8 ⊢ (𝐴 Fn 𝐶 → dom 𝐴 = 𝐶)
58	57	eqeq1d 2754	. . . . . . 7 ⊢ (𝐴 Fn 𝐶 → (dom 𝐴 = ∅ ↔ 𝐶 = ∅))
59	56, 58	bitrd 281	. . . . . 6 ⊢ (𝐴 Fn 𝐶 → (𝐴 = ∅ ↔ 𝐶 = ∅))
60	59	adantr 483	. . . . 5 ⊢ ((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) → (𝐴 = ∅ ↔ 𝐶 = ∅))
61	60	adantr 483	. . . 4 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → (𝐴 = ∅ ↔ 𝐶 = ∅))
62	47, 53, 61	3imtr4d 296	. . 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 214	1 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → (𝐴 = ∅ ↔ (𝐴 + 𝐵) = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 856   = wceq 1550   ∈ wcel 2132   ≠ wne 2947  ∃wrex 3076  Vcvv 3444   ∖ cdif 3892   ∪ cun 3893   ⊆ wss 3895  ∅c0 4276  {copab 5152  dom cdm 5636  Rel wrel 5641  Ord word 6330  Oncon0 6331   Fn wfn 6501  ‘cfv 6506  (class class class)co 7381   ∈ cmpo 7383  ωcom 7831   +_o coa 8418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703  ax-reg 9526

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-rmo 3357  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-pss 3915  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-int 4896  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-tr 5198  df-id 5531  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5589  df-we 5591  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-pred 6273  df-ord 6334  df-on 6335  df-lim 6336  df-suc 6337  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-ov 7384  df-oprab 7385  df-mpo 7386  df-om 7832  df-2nd 7956  df-frecs 8246  df-wrecs 8277  df-recs 8326  df-rdg 8365  df-oadd 8425

This theorem is referenced by: (None)