Theorem tfsconcatfv 43772

Description: The value of the concatenation of two transfinite series. (Contributed by RP, 24-Feb-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
tfsconcatfv	⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) → ((𝐴 + 𝐵)‘𝑋) = if(𝑋 ∈ 𝐶, (𝐴‘𝑋), (𝐵‘(℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋))))

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

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

Proof of Theorem tfsconcatfv

Step	Hyp	Ref	Expression
1		tfsconcat.op	. . . . 5 ⊢ + = (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)))}))
2	1	tfsconcatfv1 43770	. . . 4 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐶) → ((𝐴 + 𝐵)‘𝑋) = (𝐴‘𝑋))
3	2	adantlr 716	. . 3 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ 𝑋 ∈ 𝐶) → ((𝐴 + 𝐵)‘𝑋) = (𝐴‘𝑋))
4		simpr 484	. . . 4 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝐶)
5	4	iftrued 4475	. . 3 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ 𝑋 ∈ 𝐶) → if(𝑋 ∈ 𝐶, (𝐴‘𝑋), (𝐵‘(℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋))) = (𝐴‘𝑋))
6	3, 5	eqtr4d 2775	. 2 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ 𝑋 ∈ 𝐶) → ((𝐴 + 𝐵)‘𝑋) = if(𝑋 ∈ 𝐶, (𝐴‘𝑋), (𝐵‘(℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋))))
7		simpr 484	. . . 4 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋 ∈ 𝐶) → ¬ 𝑋 ∈ 𝐶)
8	7	iffalsed 4478	. . 3 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋 ∈ 𝐶) → if(𝑋 ∈ 𝐶, (𝐴‘𝑋), (𝐵‘(℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋))) = (𝐵‘(℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋)))
9		simpll 767	. . . 4 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋 ∈ 𝐶) → ((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)))
10		onss 7730	. . . . . . . 8 ⊢ (𝐷 ∈ On → 𝐷 ⊆ On)
11	10	adantl 481	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → 𝐷 ⊆ On)
12	11	ad3antlr 732	. . . . . 6 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋 ∈ 𝐶) → 𝐷 ⊆ On)
13		simpllr 776	. . . . . . 7 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋 ∈ 𝐶) → (𝐶 ∈ On ∧ 𝐷 ∈ On))
14		simplrl 777	. . . . . . . . 9 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) → 𝐶 ∈ On)
15		oacl 8461	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐶 +o 𝐷) ∈ On)
16	15	adantl 481	. . . . . . . . . 10 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐶 +o 𝐷) ∈ On)
17		onelon 6340	. . . . . . . . . 10 ⊢ (((𝐶 +o 𝐷) ∈ On ∧ 𝑋 ∈ (𝐶 +o 𝐷)) → 𝑋 ∈ On)
18	16, 17	sylan 581	. . . . . . . . 9 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) → 𝑋 ∈ On)
19		ontri1 6349	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ 𝑋 ∈ On) → (𝐶 ⊆ 𝑋 ↔ ¬ 𝑋 ∈ 𝐶))
20	14, 18, 19	syl2anc 585	. . . . . . . 8 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) → (𝐶 ⊆ 𝑋 ↔ ¬ 𝑋 ∈ 𝐶))
21	20	biimpar 477	. . . . . . 7 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋 ∈ 𝐶) → 𝐶 ⊆ 𝑋)
22		simplr 769	. . . . . . 7 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋 ∈ 𝐶) → 𝑋 ∈ (𝐶 +o 𝐷))
23		oawordex2 43757	. . . . . . 7 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ On) ∧ (𝐶 ⊆ 𝑋 ∧ 𝑋 ∈ (𝐶 +o 𝐷))) → ∃𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋)
24	13, 21, 22, 23	syl12anc 837	. . . . . 6 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋 ∈ 𝐶) → ∃𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋)
25	14, 18	jca 511	. . . . . . 7 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) → (𝐶 ∈ On ∧ 𝑋 ∈ On))
26		oawordeu 8481	. . . . . . 7 ⊢ (((𝐶 ∈ On ∧ 𝑋 ∈ On) ∧ 𝐶 ⊆ 𝑋) → ∃!𝑑 ∈ On (𝐶 +o 𝑑) = 𝑋)
27	25, 21, 26	syl2an2r 686	. . . . . 6 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋 ∈ 𝐶) → ∃!𝑑 ∈ On (𝐶 +o 𝑑) = 𝑋)
28		reuss 4268	. . . . . 6 ⊢ ((𝐷 ⊆ On ∧ ∃𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋 ∧ ∃!𝑑 ∈ On (𝐶 +o 𝑑) = 𝑋) → ∃!𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋)
29	12, 24, 27, 28	syl3anc 1374	. . . . 5 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋 ∈ 𝐶) → ∃!𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋)
30		riotacl 7332	. . . . 5 ⊢ (∃!𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋 → (℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋) ∈ 𝐷)
31	29, 30	syl 17	. . . 4 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋 ∈ 𝐶) → (℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋) ∈ 𝐷)
32	1	tfsconcatfv2 43771	. . . 4 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ (℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋) ∈ 𝐷) → ((𝐴 + 𝐵)‘(𝐶 +o (℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋))) = (𝐵‘(℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋)))
33	9, 31, 32	syl2anc 585	. . 3 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋 ∈ 𝐶) → ((𝐴 + 𝐵)‘(𝐶 +o (℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋))) = (𝐵‘(℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋)))
34		riotasbc 7333	. . . . . 6 ⊢ (∃!𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋 → [(℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋) / 𝑑](𝐶 +o 𝑑) = 𝑋)
35	29, 34	syl 17	. . . . 5 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋 ∈ 𝐶) → [(℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋) / 𝑑](𝐶 +o 𝑑) = 𝑋)
36		sbceq1g 4358	. . . . . . 7 ⊢ ((℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋) ∈ 𝐷 → ([(℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋) / 𝑑](𝐶 +o 𝑑) = 𝑋 ↔ ⦋(℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋) / 𝑑⦌(𝐶 +o 𝑑) = 𝑋))
37		csbov2g 7406	. . . . . . . . 9 ⊢ ((℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋) ∈ 𝐷 → ⦋(℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋) / 𝑑⦌(𝐶 +o 𝑑) = (𝐶 +o ⦋(℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋) / 𝑑⦌𝑑))
38		csbvarg 4375	. . . . . . . . . 10 ⊢ ((℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋) ∈ 𝐷 → ⦋(℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋) / 𝑑⦌𝑑 = (℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋))
39	38	oveq2d 7374	. . . . . . . . 9 ⊢ ((℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋) ∈ 𝐷 → (𝐶 +o ⦋(℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋) / 𝑑⦌𝑑) = (𝐶 +o (℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋)))
40	37, 39	eqtrd 2772	. . . . . . . 8 ⊢ ((℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋) ∈ 𝐷 → ⦋(℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋) / 𝑑⦌(𝐶 +o 𝑑) = (𝐶 +o (℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋)))
41	40	eqeq1d 2739	. . . . . . 7 ⊢ ((℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋) ∈ 𝐷 → (⦋(℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋) / 𝑑⦌(𝐶 +o 𝑑) = 𝑋 ↔ (𝐶 +o (℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋)) = 𝑋))
42	36, 41	bitrd 279	. . . . . 6 ⊢ ((℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋) ∈ 𝐷 → ([(℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋) / 𝑑](𝐶 +o 𝑑) = 𝑋 ↔ (𝐶 +o (℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋)) = 𝑋))
43	42	biimpa 476	. . . . 5 ⊢ (((℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋) ∈ 𝐷 ∧ [(℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋) / 𝑑](𝐶 +o 𝑑) = 𝑋) → (𝐶 +o (℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋)) = 𝑋)
44	31, 35, 43	syl2anc 585	. . . 4 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋 ∈ 𝐶) → (𝐶 +o (℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋)) = 𝑋)
45	44	fveq2d 6836	. . 3 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋 ∈ 𝐶) → ((𝐴 + 𝐵)‘(𝐶 +o (℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋))) = ((𝐴 + 𝐵)‘𝑋))
46	8, 33, 45	3eqtr2rd 2779	. 2 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋 ∈ 𝐶) → ((𝐴 + 𝐵)‘𝑋) = if(𝑋 ∈ 𝐶, (𝐴‘𝑋), (𝐵‘(℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋))))
47	6, 46	pm2.61dan 813	1 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) → ((𝐴 + 𝐵)‘𝑋) = if(𝑋 ∈ 𝐶, (𝐴‘𝑋), (𝐵‘(℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  ∃!wreu 3341  Vcvv 3430  [wsbc 3729  ⦋csb 3838   ∖ cdif 3887   ∪ cun 3888   ⊆ wss 3890  ifcif 4467  {copab 5148  dom cdm 5622  Oncon0 6315   Fn wfn 6485  ‘cfv 6490  ℩crio 7314  (class class class)co 7358   ∈ cmpo 7360   +_o coa 8393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-oadd 8400

This theorem is referenced by: (None)