Theorem tfsconcatfv 44016

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 44014	. . . 4 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐶) → ((𝐴 + 𝐵)‘𝑋) = (𝐴‘𝑋))
3	2	adantlr 727	. . 3 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ 𝑋 ∈ 𝐶) → ((𝐴 + 𝐵)‘𝑋) = (𝐴‘𝑋))
4		simpr 489	. . . 4 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝐶)
5	4	iftrued 4494	. . 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 2799	. 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 489	. . . 4 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋 ∈ 𝐶) → ¬ 𝑋 ∈ 𝐶)
8	7	iffalsed 4497	. . 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 778	. . . 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 7783	. . . . . . . 8 ⊢ (𝐷 ∈ On → 𝐷 ⊆ On)
11	10	adantl 486	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → 𝐷 ⊆ On)
12	11	ad3antlr 743	. . . . . 6 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋 ∈ 𝐶) → 𝐷 ⊆ On)
13		simpllr 787	. . . . . . 7 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋 ∈ 𝐶) → (𝐶 ∈ On ∧ 𝐷 ∈ On))
14		simplrl 788	. . . . . . . . 9 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) → 𝐶 ∈ On)
15		oacl 8519	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐶 +o 𝐷) ∈ On)
16	15	adantl 486	. . . . . . . . . 10 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐶 +o 𝐷) ∈ On)
17		onelon 6385	. . . . . . . . . 10 ⊢ (((𝐶 +o 𝐷) ∈ On ∧ 𝑋 ∈ (𝐶 +o 𝐷)) → 𝑋 ∈ On)
18	16, 17	sylan 591	. . . . . . . . 9 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) → 𝑋 ∈ On)
19		ontri1 6395	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ 𝑋 ∈ On) → (𝐶 ⊆ 𝑋 ↔ ¬ 𝑋 ∈ 𝐶))
20	14, 18, 19	syl2anc 595	. . . . . . . 8 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) → (𝐶 ⊆ 𝑋 ↔ ¬ 𝑋 ∈ 𝐶))
21	20	biimpar 482	. . . . . . 7 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋 ∈ 𝐶) → 𝐶 ⊆ 𝑋)
22		simplr 780	. . . . . . 7 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) ∧ ¬ 𝑋 ∈ 𝐶) → 𝑋 ∈ (𝐶 +o 𝐷))
23		oawordex2 44001	. . . . . . 7 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ On) ∧ (𝐶 ⊆ 𝑋 ∧ 𝑋 ∈ (𝐶 +o 𝐷))) → ∃𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋)
24	13, 21, 22, 23	syl12anc 849	. . . . . 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 520	. . . . . . 7 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) → (𝐶 ∈ On ∧ 𝑋 ∈ On))
26		oawordeu 8539	. . . . . . 7 ⊢ (((𝐶 ∈ On ∧ 𝑋 ∈ On) ∧ 𝐶 ⊆ 𝑋) → ∃!𝑑 ∈ On (𝐶 +o 𝑑) = 𝑋)
27	25, 21, 26	syl2an2r 697	. . . . . 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 4279	. . . . . 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 1396	. . . . 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 7384	. . . . 5 ⊢ (∃!𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋 → (℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋) ∈ 𝐷)
31	29, 30	syl 18	. . . 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 44015	. . . 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 595	. . 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 7385	. . . . . 6 ⊢ (∃!𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋 → [(℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋) / 𝑑](𝐶 +o 𝑑) = 𝑋)
35	29, 34	syl 18	. . . . 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 4381	. . . . . . 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 7458	. . . . . . . . 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 4398	. . . . . . . . . 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 7426	. . . . . . . . 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 2796	. . . . . . . 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 2763	. . . . . . 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 282	. . . . . 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 481	. . . . 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 595	. . . 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 6885	. . 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 2803	. 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 824	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 209   ∧ wa 400   = wceq 1568   ∈ wcel 2141  ∃wrex 3087  ∃!wreu 3365  Vcvv 3453  [wsbc 3743  ⦋csb 3852   ∖ cdif 3901   ∪ cun 3902   ⊆ wss 3904  ifcif 4486  {copab 5172  dom cdm 5661  Oncon0 6360   Fn wfn 6531  ‘cfv 6536  ℩crio 7366  (class class class)co 7410   ∈ cmpo 7412   +_o coa 8449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7862  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-oadd 8456

This theorem is referenced by: (None)