relexpxpmin - Mathbox for Richard Penner

	Mathbox for Richard Penner		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > relexpxpmin		Structured version Visualization version GIF version

Theorem relexpxpmin 44978

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 12648	. . . 4 ⊢ (𝐾 ∈ ℕ₀ ↔ (𝐾 ∈ ℕ ∨ 𝐾 = 0))
2		elnn0 12648	. . . . . 6 ⊢ (𝐽 ∈ ℕ₀ ↔ (𝐽 ∈ ℕ ∨ 𝐽 = 0))
3		ifeqor 4537	. . . . . . . . . 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 2783	. . . . . . . . . 10 ⊢ ((𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) ∧ if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐽) → 𝐼 = 𝐽)
8		eqtr 2783	. . . . . . . . . 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 44964	. . . . . . . . . . . . . 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 44964	. . . . . . . . . . . . . . 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 7466	. . . . . . . . . . . 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 7467	. . . . . . . . . . . . 13 ⊢ (((𝐼 = 𝐽 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐼) = ((𝐴 × 𝐵)↑_𝑟𝐽))
19	18, 15	eqtrd 2798	. . . . . . . . . . . 12 ⊢ (((𝐼 = 𝐽 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐼) = (𝐴 × 𝐵))
20	12, 16, 19	3eqtr4d 2808	. . . . . . . . . . 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 2769	. . . . . . . . . . . 12 ⊢ (((𝐼 = 𝐾 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐾 = 𝐼)
25	22, 24	oveq12d 7469	. . . . . . . . . . 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 12427	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 0 < 𝐾)
34	31, 33	eqbrtrd 5133	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐽 < 𝐾)
35	34	iftrued 4493	. . . . . . . . . 10 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐽)
36	30, 35, 31	3eqtrd 2802	. . . . . . . . 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 7797	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (𝐴 × 𝐵) ∈ V)
40		dmexg 7947	. . . . . . . . . . . . . 14 ⊢ ((𝐴 × 𝐵) ∈ V → dom (𝐴 × 𝐵) ∈ V)
41		rnexg 7948	. . . . . . . . . . . . . 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 7792	. . . . . . . . . . . . 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 12707	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐾 ∈ ℕ₀)
47		relexpiidm 44965	. . . . . . . . . . . 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 7467	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐽) = ((𝐴 × 𝐵)↑_𝑟0))
51		relexp0g 15218	. . . . . . . . . . . . . 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 2798	. . . . . . . . . . . 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 7466	. . . . . . . . . . 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 7467	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐼) = ((𝐴 × 𝐵)↑_𝑟0))
57	56, 52	eqtrd 2798	. . . . . . . . . . 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 2808	. . . . . . . . . 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 1437	. . . . . . . 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 12672	. . . . . . . . . . 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 5120	. . . . . . . . . 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 4496	. . . . . . . 8 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
73	67, 72, 64	3eqtrd 2802	. . . . . . 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 7466	. . . . . . . . . 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 7467	. . . . . . . . . 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 7467	. . . . . . . . . 10 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐼) = ((𝐴 × 𝐵)↑_𝑟0))
81	76, 78, 80	3eqtr4d 2808	. . . . . . . . 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 7797	. . . . . . . . . . 11 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (𝐴 × 𝐵) ∈ V)
86		relexp0idm 44976	. . . . . . . . . . 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 7467	. . . . . . . . . . 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 7469	. . . . . . . . . 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 7467	. . . . . . . . . 10 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐼) = ((𝐴 × 𝐵)↑_𝑟0))
94	87, 91, 93	3eqtr4d 2808	. . . . . . . . 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 2147 ≠ wne 2959 Vcvv 3454 ∩ cin 4092 ∪ cun 4152 ∅c0 4281 ifcif 4485 class class class wbr 5108 I cid 5552 × cxp 5656 dom cdm 5658 ran crn 5659 ↾ cres 5660 (class class class)co 7451 0cc0 11239 < clt 11382 ℕcn 12375 ℕ₀cn0 12646 ↑_𝑟crelexp 15215

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-er 8755 df-en 9012 df-dom 9013 df-sdom 9014 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-nn 12376 df-n0 12647 df-z 12734 df-uz 13006 df-seq 14185 df-relexp 15216

This theorem is used by: (None)

W3C validator