Theorem tfsconcatfv 43290

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 43288	. . . 4 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐶) → ((𝐴 + 𝐵)‘𝑋) = (𝐴‘𝑋))
3	2	adantlr 715	. . 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 4506	. . 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 2772	. 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 4509	. . 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 766	. . . 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 7773	. . . . . . . 8 ⊢ (𝐷 ∈ On → 𝐷 ⊆ On)
11	10	adantl 481	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → 𝐷 ⊆ On)
12	11	ad3antlr 731	. . . . . 6 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋 ∈ 𝐶) → 𝐷 ⊆ On)
13		simpllr 775	. . . . . . 7 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋 ∈ 𝐶) → (𝐶 ∈ On ∧ 𝐷 ∈ On))
14		simplrl 776	. . . . . . . . 9 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) → 𝐶 ∈ On)
15		oacl 8541	. . . . . . . . . . 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 6374	. . . . . . . . . 10 ⊢ (((𝐶 +o 𝐷) ∈ On ∧ 𝑋 ∈ (𝐶 +o 𝐷)) → 𝑋 ∈ On)
18	16, 17	sylan 580	. . . . . . . . 9 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) → 𝑋 ∈ On)
19		ontri1 6383	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ 𝑋 ∈ On) → (𝐶 ⊆ 𝑋 ↔ ¬ 𝑋 ∈ 𝐶))
20	14, 18, 19	syl2anc 584	. . . . . . . 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 768	. . . . . . 7 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋 ∈ 𝐶) → 𝑋 ∈ (𝐶 +o 𝐷))
23		oawordex2 43275	. . . . . . 7 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ On) ∧ (𝐶 ⊆ 𝑋 ∧ 𝑋 ∈ (𝐶 +o 𝐷))) → ∃𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋)
24	13, 21, 22, 23	syl12anc 836	. . . . . 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 8561	. . . . . . 7 ⊢ (((𝐶 ∈ On ∧ 𝑋 ∈ On) ∧ 𝐶 ⊆ 𝑋) → ∃!𝑑 ∈ On (𝐶 +o 𝑑) = 𝑋)
27	25, 21, 26	syl2an2r 685	. . . . . 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 4300	. . . . . 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 1372	. . . . 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 7373	. . . . 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 43289	. . . 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 584	. . 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 7374	. . . . . 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 4390	. . . . . . 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 7447	. . . . . . . . 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 4407	. . . . . . . . . 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 7415	. . . . . . . . 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 2769	. . . . . . . 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 2736	. . . . . . 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 584	. . . 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 6876	. . 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 2776	. 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 812	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 1539   ∈ wcel 2107  ∃wrex 3059  ∃!wreu 3355  Vcvv 3457  [wsbc 3763  ⦋csb 3872   ∖ cdif 3921   ∪ cun 3922   ⊆ wss 3924  ifcif 4498  {copab 5178  dom cdm 5651  Oncon0 6349   Fn wfn 6522  ‘cfv 6527  ℩crio 7355  (class class class)co 7399   ∈ cmpo 7401   +_o coa 8471

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 2706  ax-rep 5246  ax-sep 5263  ax-nul 5273  ax-pow 5332  ax-pr 5399  ax-un 7723

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 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3357  df-reu 3358  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-pss 3944  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4881  df-int 4920  df-iun 4966  df-br 5117  df-opab 5179  df-mpt 5199  df-tr 5227  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6287  df-ord 6352  df-on 6353  df-lim 6354  df-suc 6355  df-iota 6480  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7356  df-ov 7402  df-oprab 7403  df-mpo 7404  df-om 7856  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8379  df-rdg 8418  df-oadd 8478

This theorem is referenced by: (None)