Theorem tfsconcat0b 43928

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 7852	. . . . 5 ⊢ (𝐷 ∈ ω → 𝐷 ∈ On)
2	1	anim2i 626	. . . 4 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (𝐶 ∈ On ∧ 𝐷 ∈ On))
3	2	anim2i 626	. . 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 43927	. . 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 5880	. . 3 ⊢ ((𝐴 + 𝐵) = 𝐵 → dom (𝐴 + 𝐵) = dom 𝐵)
8		nna0r 8579	. . . . . . . . 9 ⊢ (𝐷 ∈ ω → (∅ +o 𝐷) = 𝐷)
9	8	adantl 485	. . . . . . . 8 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (∅ +o 𝐷) = 𝐷)
10	9	eqeq2d 2774	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → ((𝐶 +o 𝐷) = (∅ +o 𝐷) ↔ (𝐶 +o 𝐷) = 𝐷))
11		eqcom 2770	. . . . . . 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 6403	. . . . . . . . . 10 ⊢ (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
14	13	adantr 484	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
15		df-ne 2959	. . . . . . . . 9 ⊢ (𝐶 ≠ ∅ ↔ ¬ 𝐶 = ∅)
16	14, 15	bitr2di 290	. . . . . . . 8 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (¬ 𝐶 = ∅ ↔ ∅ ∈ 𝐶))
17		peano1 7869	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ ω
18		nnaordr 8590	. . . . . . . . . . . . . . 15 ⊢ ((∅ ∈ ω ∧ 𝐶 ∈ ω ∧ 𝐷 ∈ ω) → (∅ ∈ 𝐶 ↔ (∅ +o 𝐷) ∈ (𝐶 +o 𝐷)))
19	17, 18	mp3an1 1470	. . . . . . . . . . . . . 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 416	. . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . 16 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ ω) ∧ ω ⊆ 𝐶) → ω ⊆ 𝐶)
24		oaword1 8521	. . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ ω) ∧ ω ⊆ 𝐶) → 𝐶 ⊆ (𝐶 +o 𝐷))
27	23, 26	sstrd 3947	. . . . . . . . . . . . . . 15 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ ω) ∧ ω ⊆ 𝐶) → ω ⊆ (𝐶 +o 𝐷))
28		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝐷 ∈ ω → 𝐷 ∈ ω)
29	8, 28	eqeltrd 2863	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 ∈ ω → (∅ +o 𝐷) ∈ ω)
30	29	ad2antlr 737	. . . . . . . . . . . . . . 15 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ ω) ∧ ω ⊆ 𝐶) → (∅ +o 𝐷) ∈ ω)
31	27, 30	sseldd 3938	. . . . . . . . . . . . . 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 423	. . . . . . . . . . . 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 6356	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ On → Ord 𝐶)
36		ordom 7856	. . . . . . . . . . . 12 ⊢ Ord ω
37		ordtri2or 6446	. . . . . . . . . . . 12 ⊢ ((Ord 𝐶 ∧ Ord ω) → (𝐶 ∈ ω ∨ ω ⊆ 𝐶))
38	35, 36, 37	sylancl 595	. . . . . . . . . . 11 ⊢ (𝐶 ∈ On → (𝐶 ∈ ω ∨ ω ⊆ 𝐶))
39	22, 34, 38	mpjaod 871	. . . . . . . . . 10 ⊢ (𝐶 ∈ On → (𝐷 ∈ ω → (∅ ∈ 𝐶 → (∅ +o 𝐷) ∈ (𝐶 +o 𝐷))))
40	39	imp 410	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (∅ ∈ 𝐶 → (∅ +o 𝐷) ∈ (𝐶 +o 𝐷)))
41		elneq 9547	. . . . . . . . . 10 ⊢ ((∅ +o 𝐷) ∈ (𝐶 +o 𝐷) → (∅ +o 𝐷) ≠ (𝐶 +o 𝐷))
42	41	neneqd 2963	. . . . . . . . 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 485	. . . 4 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → ((𝐶 +o 𝐷) = 𝐷 → 𝐶 = ∅))
48	4	tfsconcatfn 43920	. . . . . . 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 6626	. . . . 5 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → dom (𝐴 + 𝐵) = (𝐶 +o 𝐷))
51		fndm 6624	. . . . . 6 ⊢ (𝐵 Fn 𝐷 → dom 𝐵 = 𝐷)
52	51	ad2antlr 737	. . . . 5 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → dom 𝐵 = 𝐷)
53	50, 52	eqeq12d 2779	. . . 4 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → (dom (𝐴 + 𝐵) = dom 𝐵 ↔ (𝐶 +o 𝐷) = 𝐷))
54		fnrel 6623	. . . . . . . 8 ⊢ (𝐴 Fn 𝐶 → Rel 𝐴)
55		reldm0 5905	. . . . . . . 8 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅))
56	54, 55	syl 17	. . . . . . 7 ⊢ (𝐴 Fn 𝐶 → (𝐴 = ∅ ↔ dom 𝐴 = ∅))
57		fndm 6624	. . . . . . . 8 ⊢ (𝐴 Fn 𝐶 → dom 𝐴 = 𝐶)
58	57	eqeq1d 2765	. . . . . . 7 ⊢ (𝐴 Fn 𝐶 → (dom 𝐴 = ∅ ↔ 𝐶 = ∅))
59	56, 58	bitrd 281	. . . . . 6 ⊢ (𝐴 Fn 𝐶 → (𝐴 = ∅ ↔ 𝐶 = ∅))
60	59	adantr 484	. . . . 5 ⊢ ((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) → (𝐴 = ∅ ↔ 𝐶 = ∅))
61	60	adantr 484	. . . 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 399   ∨ wo 858   = wceq 1561   ∈ wcel 2143   ≠ wne 2958  ∃wrex 3087  Vcvv 3455   ∖ cdif 3902   ∪ cun 3903   ⊆ wss 3905  ∅c0 4286  {copab 5163  dom cdm 5648  Rel wrel 5653  Ord word 6345  Oncon0 6346   Fn wfn 6516  ‘cfv 6521  (class class class)co 7396   ∈ cmpo 7398  ωcom 7846   +_o coa 8434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7718  ax-reg 9538

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-oadd 8441

This theorem is referenced by: (None)