relexpxpmin - Mathbox for Richard Penner

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Theorem relexpxpmin 40055

Description: The composition of powers of a cross-product of non-disjoint sets is the cross product raised to the minimum exponent. (Contributed by RP, 13-Jun-2020.)

**Assertion**
Ref	Expression
relexpxpmin	⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) ∧ (𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) ∧ 𝐽 ∈ ℕ₀ ∧ 𝐾 ∈ ℕ₀)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼))

**Proof of Theorem relexpxpmin**
Step	Hyp	Ref	Expression
1		elnn0 11893	. . . 4 ⊢ (𝐾 ∈ ℕ₀ ↔ (𝐾 ∈ ℕ ∨ 𝐾 = 0))
2		elnn0 11893	. . . . . 6 ⊢ (𝐽 ∈ ℕ₀ ↔ (𝐽 ∈ ℕ ∨ 𝐽 = 0))
3		ifeqor 4516	. . . . . . . . . 10 ⊢ (if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐽 ∨ if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
4		andi 1004	. . . . . . . . . . 11 ⊢ ((𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) ∧ (if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐽 ∨ if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)) ↔ ((𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) ∧ if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐽) ∨ (𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) ∧ if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)))
5	4	biimpi 218	. . . . . . . . . 10 ⊢ ((𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) ∧ (if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐽 ∨ if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)) → ((𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) ∧ if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐽) ∨ (𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) ∧ if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)))
6	3, 5	mpan2 689	. . . . . . . . 9 ⊢ (𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) → ((𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) ∧ if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐽) ∨ (𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) ∧ if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)))
7		eqtr 2841	. . . . . . . . . 10 ⊢ ((𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) ∧ if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐽) → 𝐼 = 𝐽)
8		eqtr 2841	. . . . . . . . . 10 ⊢ ((𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) ∧ if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾) → 𝐼 = 𝐾)
9	7, 8	orim12i 905	. . . . . . . . 9 ⊢ (((𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) ∧ if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐽) ∨ (𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) ∧ if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)) → (𝐼 = 𝐽 ∨ 𝐼 = 𝐾))
10		relexpxpnnidm 40041	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ ℕ → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → ((𝐴 × 𝐵)↑_𝑟𝐾) = (𝐴 × 𝐵)))
11	10	imp 409	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ ℕ ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐾) = (𝐴 × 𝐵))
12	11	3ad2antl3 1183	. . . . . . . . . . . 12 ⊢ (((𝐼 = 𝐽 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐾) = (𝐴 × 𝐵))
13		relexpxpnnidm 40041	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ ℕ → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → ((𝐴 × 𝐵)↑_𝑟𝐽) = (𝐴 × 𝐵)))
14	13	imp 409	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ ℕ ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐽) = (𝐴 × 𝐵))
15	14	3ad2antl2 1182	. . . . . . . . . . . . 13 ⊢ (((𝐼 = 𝐽 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐽) = (𝐴 × 𝐵))
16	15	oveq1d 7165	. . . . . . . . . . . 12 ⊢ (((𝐼 = 𝐽 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐾))
17		simpl1 1187	. . . . . . . . . . . . . 14 ⊢ (((𝐼 = 𝐽 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐼 = 𝐽)
18	17	oveq2d 7166	. . . . . . . . . . . . 13 ⊢ (((𝐼 = 𝐽 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐼) = ((𝐴 × 𝐵)↑_𝑟𝐽))
19	18, 15	eqtrd 2856	. . . . . . . . . . . 12 ⊢ (((𝐼 = 𝐽 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐼) = (𝐴 × 𝐵))
20	12, 16, 19	3eqtr4d 2866	. . . . . . . . . . 11 ⊢ (((𝐼 = 𝐽 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼))
21	20	3exp1 1348	. . . . . . . . . 10 ⊢ (𝐼 = 𝐽 → (𝐽 ∈ ℕ → (𝐾 ∈ ℕ → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))))
22	14	3ad2antl2 1182	. . . . . . . . . . . 12 ⊢ (((𝐼 = 𝐾 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐽) = (𝐴 × 𝐵))
23		simpl1 1187	. . . . . . . . . . . . 13 ⊢ (((𝐼 = 𝐾 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐼 = 𝐾)
24	23	eqcomd 2827	. . . . . . . . . . . 12 ⊢ (((𝐼 = 𝐾 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐾 = 𝐼)
25	22, 24	oveq12d 7168	. . . . . . . . . . 11 ⊢ (((𝐼 = 𝐾 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼))
26	25	3exp1 1348	. . . . . . . . . 10 ⊢ (𝐼 = 𝐾 → (𝐽 ∈ ℕ → (𝐾 ∈ ℕ → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))))
27	21, 26	jaoi 853	. . . . . . . . 9 ⊢ ((𝐼 = 𝐽 ∨ 𝐼 = 𝐾) → (𝐽 ∈ ℕ → (𝐾 ∈ ℕ → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))))
28	6, 9, 27	3syl 18	. . . . . . . 8 ⊢ (𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) → (𝐽 ∈ ℕ → (𝐾 ∈ ℕ → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))))
29	28	com13 88	. . . . . . 7 ⊢ (𝐾 ∈ ℕ → (𝐽 ∈ ℕ → (𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))))
30		simp3 1134	. . . . . . . . . 10 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
31		simp2 1133	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐽 = 0)
32		simp1 1132	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐾 ∈ ℕ)
33	32	nngt0d 11680	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 0 < 𝐾)
34	31, 33	eqbrtrd 5081	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐽 < 𝐾)
35	34	iftrued 4475	. . . . . . . . . 10 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐽)
36	30, 35, 31	3eqtrd 2860	. . . . . . . . 9 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐼 = 0)
37		simpr1 1190	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐴 ∈ 𝑈)
38		simpr2 1191	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐵 ∈ 𝑉)
39	37, 38	xpexd 7468	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (𝐴 × 𝐵) ∈ V)
40		dmexg 7607	. . . . . . . . . . . . . 14 ⊢ ((𝐴 × 𝐵) ∈ V → dom (𝐴 × 𝐵) ∈ V)
41		rnexg 7608	. . . . . . . . . . . . . 14 ⊢ ((𝐴 × 𝐵) ∈ V → ran (𝐴 × 𝐵) ∈ V)
42	40, 41	jca 514	. . . . . . . . . . . . 13 ⊢ ((𝐴 × 𝐵) ∈ V → (dom (𝐴 × 𝐵) ∈ V ∧ ran (𝐴 × 𝐵) ∈ V))
43		unexg 7466	. . . . . . . . . . . . 13 ⊢ ((dom (𝐴 × 𝐵) ∈ V ∧ ran (𝐴 × 𝐵) ∈ V) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) ∈ V)
44	39, 42, 43	3syl 18	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) ∈ V)
45		simpl1 1187	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐾 ∈ ℕ)
46	45	nnnn0d 11949	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐾 ∈ ℕ₀)
47		relexpiidm 40042	. . . . . . . . . . . 12 ⊢ (((dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) ∈ V ∧ 𝐾 ∈ ℕ₀) → (( I ↾ (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)))↑_𝑟𝐾) = ( I ↾ (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵))))
48	44, 46, 47	syl2anc 586	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (( I ↾ (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)))↑_𝑟𝐾) = ( I ↾ (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵))))
49		simpl2 1188	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐽 = 0)
50	49	oveq2d 7166	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐽) = ((𝐴 × 𝐵)↑_𝑟0))
51		relexp0g 14375	. . . . . . . . . . . . . 14 ⊢ ((𝐴 × 𝐵) ∈ V → ((𝐴 × 𝐵)↑_𝑟0) = ( I ↾ (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵))))
52	39, 51	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟0) = ( I ↾ (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵))))
53	50, 52	eqtrd 2856	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐽) = ( I ↾ (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵))))
54	53	oveq1d 7165	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = (( I ↾ (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)))↑_𝑟𝐾))
55		simpl3 1189	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐼 = 0)
56	55	oveq2d 7166	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐼) = ((𝐴 × 𝐵)↑_𝑟0))
57	56, 52	eqtrd 2856	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐼) = ( I ↾ (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵))))
58	48, 54, 57	3eqtr4d 2866	. . . . . . . . . 10 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼))
59	58	ex 415	. . . . . . . . 9 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))
60	36, 59	syld3an3 1405	. . . . . . . 8 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))
61	60	3exp 1115	. . . . . . 7 ⊢ (𝐾 ∈ ℕ → (𝐽 = 0 → (𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))))
62	29, 61	jaod 855	. . . . . 6 ⊢ (𝐾 ∈ ℕ → ((𝐽 ∈ ℕ ∨ 𝐽 = 0) → (𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))))
63	2, 62	syl5bi 244	. . . . 5 ⊢ (𝐾 ∈ ℕ → (𝐽 ∈ ℕ₀ → (𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))))
64		simp1 1132	. . . . . . 7 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐾 = 0)
65	2	biimpi 218	. . . . . . . 8 ⊢ (𝐽 ∈ ℕ₀ → (𝐽 ∈ ℕ ∨ 𝐽 = 0))
66	65	3ad2ant2 1130	. . . . . . 7 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → (𝐽 ∈ ℕ ∨ 𝐽 = 0))
67		simp3 1134	. . . . . . . 8 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
68		nn0nlt0 11917	. . . . . . . . . . 11 ⊢ (𝐽 ∈ ℕ₀ → ¬ 𝐽 < 0)
69	68	3ad2ant2 1130	. . . . . . . . . 10 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ¬ 𝐽 < 0)
70	64	breq2d 5071	. . . . . . . . . 10 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → (𝐽 < 𝐾 ↔ 𝐽 < 0))
71	69, 70	mtbird 327	. . . . . . . . 9 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ¬ 𝐽 < 𝐾)
72	71	iffalsed 4478	. . . . . . . 8 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
73	67, 72, 64	3eqtrd 2860	. . . . . . 7 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐼 = 0)
74	13	3ad2ant2 1130	. . . . . . . . . . . 12 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → ((𝐴 × 𝐵)↑_𝑟𝐽) = (𝐴 × 𝐵)))
75	74	imp 409	. . . . . . . . . . 11 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐽) = (𝐴 × 𝐵))
76	75	oveq1d 7165	. . . . . . . . . 10 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟0) = ((𝐴 × 𝐵)↑_𝑟0))
77		simpl1 1187	. . . . . . . . . . 11 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐾 = 0)
78	77	oveq2d 7166	. . . . . . . . . 10 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟0))
79		simpl3 1189	. . . . . . . . . . 11 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐼 = 0)
80	79	oveq2d 7166	. . . . . . . . . 10 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐼) = ((𝐴 × 𝐵)↑_𝑟0))
81	76, 78, 80	3eqtr4d 2866	. . . . . . . . 9 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼))
82	81	3exp1 1348	. . . . . . . 8 ⊢ (𝐾 = 0 → (𝐽 ∈ ℕ → (𝐼 = 0 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))))
83		simpr1 1190	. . . . . . . . . . . 12 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐴 ∈ 𝑈)
84		simpr2 1191	. . . . . . . . . . . 12 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐵 ∈ 𝑉)
85	83, 84	xpexd 7468	. . . . . . . . . . 11 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (𝐴 × 𝐵) ∈ V)
86		relexp0idm 40053	. . . . . . . . . . 11 ⊢ ((𝐴 × 𝐵) ∈ V → (((𝐴 × 𝐵)↑_𝑟0)↑_𝑟0) = ((𝐴 × 𝐵)↑_𝑟0))
87	85, 86	syl 17	. . . . . . . . . 10 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟0)↑_𝑟0) = ((𝐴 × 𝐵)↑_𝑟0))
88		simpl2 1188	. . . . . . . . . . . 12 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐽 = 0)
89	88	oveq2d 7166	. . . . . . . . . . 11 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐽) = ((𝐴 × 𝐵)↑_𝑟0))
90		simpl1 1187	. . . . . . . . . . 11 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐾 = 0)
91	89, 90	oveq12d 7168	. . . . . . . . . 10 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = (((𝐴 × 𝐵)↑_𝑟0)↑_𝑟0))
92		simpl3 1189	. . . . . . . . . . 11 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐼 = 0)
93	92	oveq2d 7166	. . . . . . . . . 10 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐼) = ((𝐴 × 𝐵)↑_𝑟0))
94	87, 91, 93	3eqtr4d 2866	. . . . . . . . 9 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼))
95	94	3exp1 1348	. . . . . . . 8 ⊢ (𝐾 = 0 → (𝐽 = 0 → (𝐼 = 0 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))))
96	82, 95	jaod 855	. . . . . . 7 ⊢ (𝐾 = 0 → ((𝐽 ∈ ℕ ∨ 𝐽 = 0) → (𝐼 = 0 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))))
97	64, 66, 73, 96	syl3c 66	. . . . . 6 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))
98	97	3exp 1115	. . . . 5 ⊢ (𝐾 = 0 → (𝐽 ∈ ℕ₀ → (𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))))
99	63, 98	jaoi 853	. . . 4 ⊢ ((𝐾 ∈ ℕ ∨ 𝐾 = 0) → (𝐽 ∈ ℕ₀ → (𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))))
100	1, 99	sylbi 219	. . 3 ⊢ (𝐾 ∈ ℕ₀ → (𝐽 ∈ ℕ₀ → (𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))))
101	100	3imp31 1108	. 2 ⊢ ((𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) ∧ 𝐽 ∈ ℕ₀ ∧ 𝐾 ∈ ℕ₀) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))
102	101	impcom 410	1 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) ∧ (𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) ∧ 𝐽 ∈ ℕ₀ ∧ 𝐾 ∈ ℕ₀)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 Vcvv 3495 ∪ cun 3934 ∩ cin 3935 ∅c0 4291 ifcif 4467 class class class wbr 5059 I cid 5454 × cxp 5548 dom cdm 5550 ran crn 5551 ↾ cres 5552 (class class class)co 7150 0cc0 10531 < clt 10669 ℕcn 11632 ℕ₀cn0 11891 ↑_𝑟crelexp 14373

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-seq 13364 df-relexp 14374

This theorem is referenced by: (None)

W3C validator