relexpxpmin - Mathbox for Richard Penner

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Theorem relexpxpmin 38009

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 11294	. . . . 5 ⊢ (𝐾 ∈ ℕ₀ ↔ (𝐾 ∈ ℕ ∨ 𝐾 = 0))
2		elnn0 11294	. . . . . . 7 ⊢ (𝐽 ∈ ℕ₀ ↔ (𝐽 ∈ ℕ ∨ 𝐽 = 0))
3		ifeqor 4132	. . . . . . . . . . 11 ⊢ (if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐽 ∨ if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
4		andi 911	. . . . . . . . . . . 12 ⊢ ((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 206	. . . . . . . . . . 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)))
6	3, 5	mpan2 707	. . . . . . . . . 10 ⊢ (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 2641	. . . . . . . . . . 11 ⊢ ((𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) ∧ if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐽) → 𝐼 = 𝐽)
8		eqtr 2641	. . . . . . . . . . 11 ⊢ ((𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) ∧ if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾) → 𝐼 = 𝐾)
9	7, 8	orim12i 538	. . . . . . . . . 10 ⊢ (((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 37995	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ ℕ → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → ((𝐴 × 𝐵)↑_𝑟𝐾) = (𝐴 × 𝐵)))
11	10	imp 445	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ ℕ ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐾) = (𝐴 × 𝐵))
12	11	3ad2antl3 1225	. . . . . . . . . . . . 13 ⊢ (((𝐼 = 𝐽 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐾) = (𝐴 × 𝐵))
13		relexpxpnnidm 37995	. . . . . . . . . . . . . . . 16 ⊢ (𝐽 ∈ ℕ → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → ((𝐴 × 𝐵)↑_𝑟𝐽) = (𝐴 × 𝐵)))
14	13	imp 445	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ ℕ ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐽) = (𝐴 × 𝐵))
15	14	3ad2antl2 1224	. . . . . . . . . . . . . 14 ⊢ (((𝐼 = 𝐽 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐽) = (𝐴 × 𝐵))
16	15	oveq1d 6665	. . . . . . . . . . . . 13 ⊢ (((𝐼 = 𝐽 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐾))
17		simpl1 1064	. . . . . . . . . . . . . . 15 ⊢ (((𝐼 = 𝐽 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐼 = 𝐽)
18	17	oveq2d 6666	. . . . . . . . . . . . . 14 ⊢ (((𝐼 = 𝐽 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐼) = ((𝐴 × 𝐵)↑_𝑟𝐽))
19	18, 15	eqtrd 2656	. . . . . . . . . . . . 13 ⊢ (((𝐼 = 𝐽 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐼) = (𝐴 × 𝐵))
20	12, 16, 19	3eqtr4d 2666	. . . . . . . . . . . 12 ⊢ (((𝐼 = 𝐽 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼))
21	20	3exp1 1283	. . . . . . . . . . 11 ⊢ (𝐼 = 𝐽 → (𝐽 ∈ ℕ → (𝐾 ∈ ℕ → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))))
22	14	3ad2antl2 1224	. . . . . . . . . . . . 13 ⊢ (((𝐼 = 𝐾 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐽) = (𝐴 × 𝐵))
23		simpl1 1064	. . . . . . . . . . . . . 14 ⊢ (((𝐼 = 𝐾 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐼 = 𝐾)
24	23	eqcomd 2628	. . . . . . . . . . . . 13 ⊢ (((𝐼 = 𝐾 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐾 = 𝐼)
25	22, 24	oveq12d 6668	. . . . . . . . . . . 12 ⊢ (((𝐼 = 𝐾 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼))
26	25	3exp1 1283	. . . . . . . . . . 11 ⊢ (𝐼 = 𝐾 → (𝐽 ∈ ℕ → (𝐾 ∈ ℕ → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))))
27	21, 26	jaoi 394	. . . . . . . . . 10 ⊢ ((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	. . . . . . . . 9 ⊢ (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	. . . . . . . 8 ⊢ (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 1063	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
31		simp2 1062	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐽 = 0)
32		simp1 1061	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐾 ∈ ℕ)
33	32	nngt0d 11064	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 0 < 𝐾)
34	31, 33	eqbrtrd 4675	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐽 < 𝐾)
35	34	iftrued 4094	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐽)
36	30, 35, 31	3eqtrd 2660	. . . . . . . . . 10 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐼 = 0)
37		simpr1 1067	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐴 ∈ 𝑈)
38		simpr2 1068	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐵 ∈ 𝑉)
39		xpexg 6960	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → (𝐴 × 𝐵) ∈ V)
40	37, 38, 39	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (𝐴 × 𝐵) ∈ V)
41		dmexg 7097	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 × 𝐵) ∈ V → dom (𝐴 × 𝐵) ∈ V)
42		rnexg 7098	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 × 𝐵) ∈ V → ran (𝐴 × 𝐵) ∈ V)
43	41, 42	jca 554	. . . . . . . . . . . . . 14 ⊢ ((𝐴 × 𝐵) ∈ V → (dom (𝐴 × 𝐵) ∈ V ∧ ran (𝐴 × 𝐵) ∈ V))
44		unexg 6959	. . . . . . . . . . . . . 14 ⊢ ((dom (𝐴 × 𝐵) ∈ V ∧ ran (𝐴 × 𝐵) ∈ V) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) ∈ V)
45	40, 43, 44	3syl 18	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) ∈ V)
46		simpl1 1064	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐾 ∈ ℕ)
47	46	nnnn0d 11351	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐾 ∈ ℕ₀)
48		relexpiidm 37996	. . . . . . . . . . . . 13 ⊢ (((dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) ∈ V ∧ 𝐾 ∈ ℕ₀) → (( I ↾ (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)))↑_𝑟𝐾) = ( I ↾ (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵))))
49	45, 47, 48	syl2anc 693	. . . . . . . . . . . 12 ⊢ (((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))))
50		simpl2 1065	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐽 = 0)
51	50	oveq2d 6666	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐽) = ((𝐴 × 𝐵)↑_𝑟0))
52		relexp0g 13762	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 × 𝐵) ∈ V → ((𝐴 × 𝐵)↑_𝑟0) = ( I ↾ (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵))))
53	40, 52	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟0) = ( I ↾ (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵))))
54	51, 53	eqtrd 2656	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐽) = ( I ↾ (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵))))
55	54	oveq1d 6665	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = (( I ↾ (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)))↑_𝑟𝐾))
56		simpl3 1066	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐼 = 0)
57	56	oveq2d 6666	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐼) = ((𝐴 × 𝐵)↑_𝑟0))
58	57, 53	eqtrd 2656	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐼) = ( I ↾ (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵))))
59	49, 55, 58	3eqtr4d 2666	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼))
60	59	ex 450	. . . . . . . . . 10 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))
61	36, 60	syld3an3 1371	. . . . . . . . 9 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))
62	61	3exp 1264	. . . . . . . 8 ⊢ (𝐾 ∈ ℕ → (𝐽 = 0 → (𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))))
63	29, 62	jaod 395	. . . . . . 7 ⊢ (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)))))
64	2, 63	syl5bi 232	. . . . . 6 ⊢ (𝐾 ∈ ℕ → (𝐽 ∈ ℕ₀ → (𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))))
65		simp1 1061	. . . . . . . 8 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐾 = 0)
66	2	biimpi 206	. . . . . . . . 9 ⊢ (𝐽 ∈ ℕ₀ → (𝐽 ∈ ℕ ∨ 𝐽 = 0))
67	66	3ad2ant2 1083	. . . . . . . 8 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → (𝐽 ∈ ℕ ∨ 𝐽 = 0))
68		simp3 1063	. . . . . . . . 9 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
69		nn0nlt0 11319	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ ℕ₀ → ¬ 𝐽 < 0)
70	69	3ad2ant2 1083	. . . . . . . . . . 11 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ¬ 𝐽 < 0)
71	65	breq2d 4665	. . . . . . . . . . 11 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → (𝐽 < 𝐾 ↔ 𝐽 < 0))
72	70, 71	mtbird 315	. . . . . . . . . 10 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ¬ 𝐽 < 𝐾)
73	72	iffalsed 4097	. . . . . . . . 9 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
74	68, 73, 65	3eqtrd 2660	. . . . . . . 8 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐼 = 0)
75	13	3ad2ant2 1083	. . . . . . . . . . . . 13 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → ((𝐴 × 𝐵)↑_𝑟𝐽) = (𝐴 × 𝐵)))
76	75	imp 445	. . . . . . . . . . . 12 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐽) = (𝐴 × 𝐵))
77	76	oveq1d 6665	. . . . . . . . . . 11 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟0) = ((𝐴 × 𝐵)↑_𝑟0))
78		simpl1 1064	. . . . . . . . . . . 12 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐾 = 0)
79	78	oveq2d 6666	. . . . . . . . . . 11 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟0))
80		simpl3 1066	. . . . . . . . . . . 12 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐼 = 0)
81	80	oveq2d 6666	. . . . . . . . . . 11 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐼) = ((𝐴 × 𝐵)↑_𝑟0))
82	77, 79, 81	3eqtr4d 2666	. . . . . . . . . 10 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼))
83	82	3exp1 1283	. . . . . . . . 9 ⊢ (𝐾 = 0 → (𝐽 ∈ ℕ → (𝐼 = 0 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))))
84		simpr1 1067	. . . . . . . . . . . . 13 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐴 ∈ 𝑈)
85		simpr2 1068	. . . . . . . . . . . . 13 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐵 ∈ 𝑉)
86	84, 85, 39	syl2anc 693	. . . . . . . . . . . 12 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (𝐴 × 𝐵) ∈ V)
87		relexp0idm 38007	. . . . . . . . . . . 12 ⊢ ((𝐴 × 𝐵) ∈ V → (((𝐴 × 𝐵)↑_𝑟0)↑_𝑟0) = ((𝐴 × 𝐵)↑_𝑟0))
88	86, 87	syl 17	. . . . . . . . . . 11 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟0)↑_𝑟0) = ((𝐴 × 𝐵)↑_𝑟0))
89		simpl2 1065	. . . . . . . . . . . . 13 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐽 = 0)
90	89	oveq2d 6666	. . . . . . . . . . . 12 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐽) = ((𝐴 × 𝐵)↑_𝑟0))
91		simpl1 1064	. . . . . . . . . . . 12 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐾 = 0)
92	90, 91	oveq12d 6668	. . . . . . . . . . 11 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = (((𝐴 × 𝐵)↑_𝑟0)↑_𝑟0))
93		simpl3 1066	. . . . . . . . . . . 12 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐼 = 0)
94	93	oveq2d 6666	. . . . . . . . . . 11 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐼) = ((𝐴 × 𝐵)↑_𝑟0))
95	88, 92, 94	3eqtr4d 2666	. . . . . . . . . 10 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼))
96	95	3exp1 1283	. . . . . . . . 9 ⊢ (𝐾 = 0 → (𝐽 = 0 → (𝐼 = 0 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))))
97	83, 96	jaod 395	. . . . . . . 8 ⊢ (𝐾 = 0 → ((𝐽 ∈ ℕ ∨ 𝐽 = 0) → (𝐼 = 0 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))))
98	65, 67, 74, 97	syl3c 66	. . . . . . 7 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))
99	98	3exp 1264	. . . . . 6 ⊢ (𝐾 = 0 → (𝐽 ∈ ℕ₀ → (𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))))
100	64, 99	jaoi 394	. . . . 5 ⊢ ((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)))))
101	1, 100	sylbi 207	. . . 4 ⊢ (𝐾 ∈ ℕ₀ → (𝐽 ∈ ℕ₀ → (𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))))
102	101	com13 88	. . 3 ⊢ (𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) → (𝐽 ∈ ℕ₀ → (𝐾 ∈ ℕ₀ → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))))
103	102	3imp 1256	. 2 ⊢ ((𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) ∧ 𝐽 ∈ ℕ₀ ∧ 𝐾 ∈ ℕ₀) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))
104	103	impcom 446	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 ∨ wo 383 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ∪ cun 3572 ∩ cin 3573 ∅c0 3915 ifcif 4086 class class class wbr 4653 I cid 5023 × cxp 5112 dom cdm 5114 ran crn 5115 ↾ cres 5116 (class class class)co 6650 0cc0 9936 < clt 10074 ℕcn 11020 ℕ₀cn0 11292 ↑_𝑟crelexp 13760

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-seq 12802 df-relexp 13761

This theorem is referenced by: (None)

W3C validator