relexpxpmin - Mathbox for Richard Penner

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Theorem relexpxpmin 44503

Description: The composition of powers of a Cartesian product of non-disjoint sets is the Cartesian 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 12523	. . . 4 ⊢ (𝐾 ∈ ℕ₀ ↔ (𝐾 ∈ ℕ ∨ 𝐾 = 0))
2		elnn0 12523	. . . . . 6 ⊢ (𝐽 ∈ ℕ₀ ↔ (𝐽 ∈ ℕ ∨ 𝐽 = 0))
3		ifeqor 4541	. . . . . . . . . 10 ⊢ (if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐽 ∨ if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
4		andi 1025	. . . . . . . . . . 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 219	. . . . . . . . . 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 704	. . . . . . . . 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 2785	. . . . . . . . . 10 ⊢ ((𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) ∧ if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐽) → 𝐼 = 𝐽)
8		eqtr 2785	. . . . . . . . . 10 ⊢ ((𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) ∧ if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾) → 𝐼 = 𝐾)
9	7, 8	orim12i 922	. . . . . . . . 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 44489	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ ℕ → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → ((𝐴 × 𝐵)↑_𝑟𝐾) = (𝐴 × 𝐵)))
11	10	imp 412	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ ℕ ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐾) = (𝐴 × 𝐵))
12	11	3ad2antl3 1206	. . . . . . . . . . . 12 ⊢ (((𝐼 = 𝐽 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐾) = (𝐴 × 𝐵))
13		relexpxpnnidm 44489	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ ℕ → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → ((𝐴 × 𝐵)↑_𝑟𝐽) = (𝐴 × 𝐵)))
14	13	imp 412	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ ℕ ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐽) = (𝐴 × 𝐵))
15	14	3ad2antl2 1205	. . . . . . . . . . . . 13 ⊢ (((𝐼 = 𝐽 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐽) = (𝐴 × 𝐵))
16	15	oveq1d 7434	. . . . . . . . . . . 12 ⊢ (((𝐼 = 𝐽 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐾))
17		simpl1 1210	. . . . . . . . . . . . . 14 ⊢ (((𝐼 = 𝐽 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐼 = 𝐽)
18	17	oveq2d 7435	. . . . . . . . . . . . 13 ⊢ (((𝐼 = 𝐽 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐼) = ((𝐴 × 𝐵)↑_𝑟𝐽))
19	18, 15	eqtrd 2800	. . . . . . . . . . . 12 ⊢ (((𝐼 = 𝐽 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐼) = (𝐴 × 𝐵))
20	12, 16, 19	3eqtr4d 2810	. . . . . . . . . . 11 ⊢ (((𝐼 = 𝐽 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼))
21	20	3exp1 1371	. . . . . . . . . 10 ⊢ (𝐼 = 𝐽 → (𝐽 ∈ ℕ → (𝐾 ∈ ℕ → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))))
22	14	3ad2antl2 1205	. . . . . . . . . . . 12 ⊢ (((𝐼 = 𝐾 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐽) = (𝐴 × 𝐵))
23		simpl1 1210	. . . . . . . . . . . . 13 ⊢ (((𝐼 = 𝐾 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐼 = 𝐾)
24	23	eqcomd 2771	. . . . . . . . . . . 12 ⊢ (((𝐼 = 𝐾 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐾 = 𝐼)
25	22, 24	oveq12d 7437	. . . . . . . . . . 11 ⊢ (((𝐼 = 𝐾 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼))
26	25	3exp1 1371	. . . . . . . . . 10 ⊢ (𝐼 = 𝐾 → (𝐽 ∈ ℕ → (𝐾 ∈ ℕ → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))))
27	21, 26	jaoi 871	. . . . . . . . 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 19	. . . . . . . 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 89	. . . . . . 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 1156	. . . . . . . . . 10 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
31		simp2 1155	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐽 = 0)
32		simp1 1154	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐾 ∈ ℕ)
33	32	nngt0d 12302	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 0 < 𝐾)
34	31, 33	eqbrtrd 5135	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐽 < 𝐾)
35	34	iftrued 4497	. . . . . . . . . 10 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐽)
36	30, 35, 31	3eqtrd 2804	. . . . . . . . 9 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐼 = 0)
37		simpr1 1213	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐴 ∈ 𝑈)
38		simpr2 1214	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐵 ∈ 𝑉)
39	37, 38	xpexd 7756	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (𝐴 × 𝐵) ∈ V)
40		dmexg 7904	. . . . . . . . . . . . . 14 ⊢ ((𝐴 × 𝐵) ∈ V → dom (𝐴 × 𝐵) ∈ V)
41		rnexg 7905	. . . . . . . . . . . . . 14 ⊢ ((𝐴 × 𝐵) ∈ V → ran (𝐴 × 𝐵) ∈ V)
42	40, 41	jca 521	. . . . . . . . . . . . 13 ⊢ ((𝐴 × 𝐵) ∈ V → (dom (𝐴 × 𝐵) ∈ V ∧ ran (𝐴 × 𝐵) ∈ V))
43		unexg 7751	. . . . . . . . . . . . 13 ⊢ ((dom (𝐴 × 𝐵) ∈ V ∧ ran (𝐴 × 𝐵) ∈ V) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) ∈ V)
44	39, 42, 43	3syl 19	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) ∈ V)
45		simpl1 1210	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐾 ∈ ℕ)
46	45	nnnn0d 12582	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐾 ∈ ℕ₀)
47		relexpiidm 44490	. . . . . . . . . . . 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 596	. . . . . . . . . . 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 1211	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐽 = 0)
50	49	oveq2d 7435	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐽) = ((𝐴 × 𝐵)↑_𝑟0))
51		relexp0g 15085	. . . . . . . . . . . . . 14 ⊢ ((𝐴 × 𝐵) ∈ V → ((𝐴 × 𝐵)↑_𝑟0) = ( I ↾ (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵))))
52	39, 51	syl 18	. . . . . . . . . . . . 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 2800	. . . . . . . . . . . 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 7434	. . . . . . . . . . 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 1212	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐼 = 0)
56	55	oveq2d 7435	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐼) = ((𝐴 × 𝐵)↑_𝑟0))
57	56, 52	eqtrd 2800	. . . . . . . . . . 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 2810	. . . . . . . . . 10 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼))
59	58	ex 418	. . . . . . . . 9 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))
60	36, 59	syld3an3 1436	. . . . . . . 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 1137	. . . . . . 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 873	. . . . . 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	biimtrid 245	. . . . 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 1154	. . . . . . 7 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐾 = 0)
65	2	biimpi 219	. . . . . . . 8 ⊢ (𝐽 ∈ ℕ₀ → (𝐽 ∈ ℕ ∨ 𝐽 = 0))
66	65	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → (𝐽 ∈ ℕ ∨ 𝐽 = 0))
67		simp3 1156	. . . . . . . 8 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
68		nn0nlt0 12547	. . . . . . . . . . 11 ⊢ (𝐽 ∈ ℕ₀ → ¬ 𝐽 < 0)
69	68	3ad2ant2 1152	. . . . . . . . . 10 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ¬ 𝐽 < 0)
70	64	breq2d 5123	. . . . . . . . . 10 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → (𝐽 < 𝐾 ↔ 𝐽 < 0))
71	69, 70	mtbird 328	. . . . . . . . 9 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ¬ 𝐽 < 𝐾)
72	71	iffalsed 4500	. . . . . . . 8 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
73	67, 72, 64	3eqtrd 2804	. . . . . . 7 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐼 = 0)
74	13	3ad2ant2 1152	. . . . . . . . . . . 12 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → ((𝐴 × 𝐵)↑_𝑟𝐽) = (𝐴 × 𝐵)))
75	74	imp 412	. . . . . . . . . . 11 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐽) = (𝐴 × 𝐵))
76	75	oveq1d 7434	. . . . . . . . . 10 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟0) = ((𝐴 × 𝐵)↑_𝑟0))
77		simpl1 1210	. . . . . . . . . . 11 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐾 = 0)
78	77	oveq2d 7435	. . . . . . . . . 10 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟0))
79		simpl3 1212	. . . . . . . . . . 11 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐼 = 0)
80	79	oveq2d 7435	. . . . . . . . . 10 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐼) = ((𝐴 × 𝐵)↑_𝑟0))
81	76, 78, 80	3eqtr4d 2810	. . . . . . . . 9 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼))
82	81	3exp1 1371	. . . . . . . 8 ⊢ (𝐾 = 0 → (𝐽 ∈ ℕ → (𝐼 = 0 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))))
83		simpr1 1213	. . . . . . . . . . . 12 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐴 ∈ 𝑈)
84		simpr2 1214	. . . . . . . . . . . 12 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐵 ∈ 𝑉)
85	83, 84	xpexd 7756	. . . . . . . . . . 11 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (𝐴 × 𝐵) ∈ V)
86		relexp0idm 44501	. . . . . . . . . . 11 ⊢ ((𝐴 × 𝐵) ∈ V → (((𝐴 × 𝐵)↑_𝑟0)↑_𝑟0) = ((𝐴 × 𝐵)↑_𝑟0))
87	85, 86	syl 18	. . . . . . . . . 10 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟0)↑_𝑟0) = ((𝐴 × 𝐵)↑_𝑟0))
88		simpl2 1211	. . . . . . . . . . . 12 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐽 = 0)
89	88	oveq2d 7435	. . . . . . . . . . 11 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐽) = ((𝐴 × 𝐵)↑_𝑟0))
90		simpl1 1210	. . . . . . . . . . 11 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐾 = 0)
91	89, 90	oveq12d 7437	. . . . . . . . . 10 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = (((𝐴 × 𝐵)↑_𝑟0)↑_𝑟0))
92		simpl3 1212	. . . . . . . . . . 11 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐼 = 0)
93	92	oveq2d 7435	. . . . . . . . . 10 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐼) = ((𝐴 × 𝐵)↑_𝑟0))
94	87, 91, 93	3eqtr4d 2810	. . . . . . . . 9 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼))
95	94	3exp1 1371	. . . . . . . 8 ⊢ (𝐾 = 0 → (𝐽 = 0 → (𝐼 = 0 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))))
96	82, 95	jaod 873	. . . . . . 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 67	. . . . . 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 1137	. . . . 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 871	. . . 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 220	. . 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 1129	. 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 413	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 Vcvv 3457 ∪ cun 3904 ∩ cin 3905 ∅c0 4286 ifcif 4489 class class class wbr 5111 I cid 5557 × cxp 5661 dom cdm 5663 ran crn 5664 ↾ cres 5665 (class class class)co 7419 0cc0 11117 < clt 11260 ℕcn 12250 ℕ₀cn0 12521 ↑_𝑟crelexp 15082

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-er 8700 df-en 8950 df-dom 8951 df-sdom 8952 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-nn 12251 df-n0 12522 df-z 12609 df-uz 12881 df-seq 14058 df-relexp 15083

This theorem is used by: (None)

W3C validator