Theorem tfsconcatfv 43330

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 43328	. . . 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 4496	. . 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 2767	. 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 4499	. . 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 7761	. . . . . . . 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 8499	. . . . . . . . . . 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 6357	. . . . . . . . . 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 6366	. . . . . . . . 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 43315	. . . . . . 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 8519	. . . . . . 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 4290	. . . . . 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 1373	. . . . 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 7361	. . . . 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 43329	. . . 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 7362	. . . . . 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 4380	. . . . . . 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 7435	. . . . . . . . 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 4397	. . . . . . . . . 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 7403	. . . . . . . . 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 2764	. . . . . . . 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 2731	. . . . . . 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 6862	. . 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 2771	. 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 1540   ∈ wcel 2109  ∃wrex 3053  ∃!wreu 3352  Vcvv 3447  [wsbc 3753  ⦋csb 3862   ∖ cdif 3911   ∪ cun 3912   ⊆ wss 3914  ifcif 4488  {copab 5169  dom cdm 5638  Oncon0 6332   Fn wfn 6506  ‘cfv 6511  ℩crio 7343  (class class class)co 7387   ∈ cmpo 7389   +_o coa 8431

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 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-oadd 8438

This theorem is referenced by: (None)