Theorem tfsconcat0b 43342

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 7851	. . . . 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 43341	. . 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 5870	. . 3 ⊢ ((𝐴 + 𝐵) = 𝐵 → dom (𝐴 + 𝐵) = dom 𝐵)
8		nna0r 8576	. . . . . . . . 9 ⊢ (𝐷 ∈ ω → (∅ +o 𝐷) = 𝐷)
9	8	adantl 481	. . . . . . . 8 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (∅ +o 𝐷) = 𝐷)
10	9	eqeq2d 2741	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → ((𝐶 +o 𝐷) = (∅ +o 𝐷) ↔ (𝐶 +o 𝐷) = 𝐷))
11		eqcom 2737	. . . . . . 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 6392	. . . . . . . . . 10 ⊢ (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
14	13	adantr 480	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
15		df-ne 2927	. . . . . . . . 9 ⊢ (𝐶 ≠ ∅ ↔ ¬ 𝐶 = ∅)
16	14, 15	bitr2di 288	. . . . . . . 8 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (¬ 𝐶 = ∅ ↔ ∅ ∈ 𝐶))
17		peano1 7868	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ ω
18		nnaordr 8587	. . . . . . . . . . . . . . 15 ⊢ ((∅ ∈ ω ∧ 𝐶 ∈ ω ∧ 𝐷 ∈ ω) → (∅ ∈ 𝐶 ↔ (∅ +o 𝐷) ∈ (𝐶 +o 𝐷)))
19	17, 18	mp3an1 1450	. . . . . . . . . . . . . 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 8519	. . . . . . . . . . . . . . . . . 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 3960	. . . . . . . . . . . . . . 15 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ ω) ∧ ω ⊆ 𝐶) → ω ⊆ (𝐶 +o 𝐷))
28		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝐷 ∈ ω → 𝐷 ∈ ω)
29	8, 28	eqeltrd 2829	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 ∈ ω → (∅ +o 𝐷) ∈ ω)
30	29	ad2antlr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ ω) ∧ ω ⊆ 𝐶) → (∅ +o 𝐷) ∈ ω)
31	27, 30	sseldd 3950	. . . . . . . . . . . . . 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 6345	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ On → Ord 𝐶)
36		ordom 7855	. . . . . . . . . . . 12 ⊢ Ord ω
37		ordtri2or 6435	. . . . . . . . . . . 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 9558	. . . . . . . . . 10 ⊢ ((∅ +o 𝐷) ∈ (𝐶 +o 𝐷) → (∅ +o 𝐷) ≠ (𝐶 +o 𝐷))
42	41	neneqd 2931	. . . . . . . . 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 43334	. . . . . . 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 727	. . . . 5 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → dom 𝐵 = 𝐷)
53	50, 52	eqeq12d 2746	. . . 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 5894	. . . . . . . 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 2732	. . . . . . 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 1540   ∈ wcel 2109   ≠ wne 2926  ∃wrex 3054  Vcvv 3450   ∖ cdif 3914   ∪ cun 3915   ⊆ wss 3917  ∅c0 4299  {copab 5172  dom cdm 5641  Rel wrel 5646  Ord word 6334  Oncon0 6335   Fn wfn 6509  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392  ωcom 7845   +_o coa 8434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-reg 9552

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-oadd 8441

This theorem is referenced by: (None)