relexpxpmin - Mathbox for Richard Penner

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Theorem relexpxpmin 43694

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 12386	. . . 4 ⊢ (𝐾 ∈ ℕ₀ ↔ (𝐾 ∈ ℕ ∨ 𝐾 = 0))
2		elnn0 12386	. . . . . 6 ⊢ (𝐽 ∈ ℕ₀ ↔ (𝐽 ∈ ℕ ∨ 𝐽 = 0))
3		ifeqor 4528	. . . . . . . . . 10 ⊢ (if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐽 ∨ if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
4		andi 1009	. . . . . . . . . . 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 216	. . . . . . . . . 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 691	. . . . . . . . 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 2749	. . . . . . . . . 10 ⊢ ((𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) ∧ if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐽) → 𝐼 = 𝐽)
8		eqtr 2749	. . . . . . . . . 10 ⊢ ((𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) ∧ if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾) → 𝐼 = 𝐾)
9	7, 8	orim12i 908	. . . . . . . . 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 43680	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ ℕ → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → ((𝐴 × 𝐵)↑_𝑟𝐾) = (𝐴 × 𝐵)))
11	10	imp 406	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ ℕ ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐾) = (𝐴 × 𝐵))
12	11	3ad2antl3 1188	. . . . . . . . . . . 12 ⊢ (((𝐼 = 𝐽 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐾) = (𝐴 × 𝐵))
13		relexpxpnnidm 43680	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ ℕ → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → ((𝐴 × 𝐵)↑_𝑟𝐽) = (𝐴 × 𝐵)))
14	13	imp 406	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ ℕ ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐽) = (𝐴 × 𝐵))
15	14	3ad2antl2 1187	. . . . . . . . . . . . 13 ⊢ (((𝐼 = 𝐽 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐽) = (𝐴 × 𝐵))
16	15	oveq1d 7364	. . . . . . . . . . . 12 ⊢ (((𝐼 = 𝐽 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐾))
17		simpl1 1192	. . . . . . . . . . . . . 14 ⊢ (((𝐼 = 𝐽 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐼 = 𝐽)
18	17	oveq2d 7365	. . . . . . . . . . . . 13 ⊢ (((𝐼 = 𝐽 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐼) = ((𝐴 × 𝐵)↑_𝑟𝐽))
19	18, 15	eqtrd 2764	. . . . . . . . . . . 12 ⊢ (((𝐼 = 𝐽 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐼) = (𝐴 × 𝐵))
20	12, 16, 19	3eqtr4d 2774	. . . . . . . . . . 11 ⊢ (((𝐼 = 𝐽 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼))
21	20	3exp1 1353	. . . . . . . . . 10 ⊢ (𝐼 = 𝐽 → (𝐽 ∈ ℕ → (𝐾 ∈ ℕ → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))))
22	14	3ad2antl2 1187	. . . . . . . . . . . 12 ⊢ (((𝐼 = 𝐾 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐽) = (𝐴 × 𝐵))
23		simpl1 1192	. . . . . . . . . . . . 13 ⊢ (((𝐼 = 𝐾 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐼 = 𝐾)
24	23	eqcomd 2735	. . . . . . . . . . . 12 ⊢ (((𝐼 = 𝐾 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐾 = 𝐼)
25	22, 24	oveq12d 7367	. . . . . . . . . . 11 ⊢ (((𝐼 = 𝐾 ∧ 𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼))
26	25	3exp1 1353	. . . . . . . . . 10 ⊢ (𝐼 = 𝐾 → (𝐽 ∈ ℕ → (𝐾 ∈ ℕ → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))))
27	21, 26	jaoi 857	. . . . . . . . 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 1138	. . . . . . . . . 10 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
31		simp2 1137	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐽 = 0)
32		simp1 1136	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐾 ∈ ℕ)
33	32	nngt0d 12177	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 0 < 𝐾)
34	31, 33	eqbrtrd 5114	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐽 < 𝐾)
35	34	iftrued 4484	. . . . . . . . . 10 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐽)
36	30, 35, 31	3eqtrd 2768	. . . . . . . . 9 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐼 = 0)
37		simpr1 1195	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐴 ∈ 𝑈)
38		simpr2 1196	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐵 ∈ 𝑉)
39	37, 38	xpexd 7687	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (𝐴 × 𝐵) ∈ V)
40		dmexg 7834	. . . . . . . . . . . . . 14 ⊢ ((𝐴 × 𝐵) ∈ V → dom (𝐴 × 𝐵) ∈ V)
41		rnexg 7835	. . . . . . . . . . . . . 14 ⊢ ((𝐴 × 𝐵) ∈ V → ran (𝐴 × 𝐵) ∈ V)
42	40, 41	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝐴 × 𝐵) ∈ V → (dom (𝐴 × 𝐵) ∈ V ∧ ran (𝐴 × 𝐵) ∈ V))
43		unexg 7679	. . . . . . . . . . . . 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 1192	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐾 ∈ ℕ)
46	45	nnnn0d 12445	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐾 ∈ ℕ₀)
47		relexpiidm 43681	. . . . . . . . . . . 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 584	. . . . . . . . . . 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 1193	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐽 = 0)
50	49	oveq2d 7365	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐽) = ((𝐴 × 𝐵)↑_𝑟0))
51		relexp0g 14929	. . . . . . . . . . . . . 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 2764	. . . . . . . . . . . 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 7364	. . . . . . . . . . 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 1194	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐼 = 0)
56	55	oveq2d 7365	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐼) = ((𝐴 × 𝐵)↑_𝑟0))
57	56, 52	eqtrd 2764	. . . . . . . . . . 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 2774	. . . . . . . . . 10 ⊢ (((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼))
59	58	ex 412	. . . . . . . . 9 ⊢ ((𝐾 ∈ ℕ ∧ 𝐽 = 0 ∧ 𝐼 = 0) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))
60	36, 59	syld3an3 1411	. . . . . . . 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 1119	. . . . . . 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 859	. . . . . 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 242	. . . . 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 1136	. . . . . . 7 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐾 = 0)
65	2	biimpi 216	. . . . . . . 8 ⊢ (𝐽 ∈ ℕ₀ → (𝐽 ∈ ℕ ∨ 𝐽 = 0))
66	65	3ad2ant2 1134	. . . . . . 7 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → (𝐽 ∈ ℕ ∨ 𝐽 = 0))
67		simp3 1138	. . . . . . . 8 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
68		nn0nlt0 12410	. . . . . . . . . . 11 ⊢ (𝐽 ∈ ℕ₀ → ¬ 𝐽 < 0)
69	68	3ad2ant2 1134	. . . . . . . . . 10 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ¬ 𝐽 < 0)
70	64	breq2d 5104	. . . . . . . . . 10 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → (𝐽 < 𝐾 ↔ 𝐽 < 0))
71	69, 70	mtbird 325	. . . . . . . . 9 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ¬ 𝐽 < 𝐾)
72	71	iffalsed 4487	. . . . . . . 8 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
73	67, 72, 64	3eqtrd 2768	. . . . . . 7 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ₀ ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → 𝐼 = 0)
74	13	3ad2ant2 1134	. . . . . . . . . . . 12 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → ((𝐴 × 𝐵)↑_𝑟𝐽) = (𝐴 × 𝐵)))
75	74	imp 406	. . . . . . . . . . 11 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐽) = (𝐴 × 𝐵))
76	75	oveq1d 7364	. . . . . . . . . 10 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟0) = ((𝐴 × 𝐵)↑_𝑟0))
77		simpl1 1192	. . . . . . . . . . 11 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐾 = 0)
78	77	oveq2d 7365	. . . . . . . . . 10 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟0))
79		simpl3 1194	. . . . . . . . . . 11 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐼 = 0)
80	79	oveq2d 7365	. . . . . . . . . 10 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐼) = ((𝐴 × 𝐵)↑_𝑟0))
81	76, 78, 80	3eqtr4d 2774	. . . . . . . . 9 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼))
82	81	3exp1 1353	. . . . . . . 8 ⊢ (𝐾 = 0 → (𝐽 ∈ ℕ → (𝐼 = 0 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))))
83		simpr1 1195	. . . . . . . . . . . 12 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐴 ∈ 𝑈)
84		simpr2 1196	. . . . . . . . . . . 12 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐵 ∈ 𝑉)
85	83, 84	xpexd 7687	. . . . . . . . . . 11 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (𝐴 × 𝐵) ∈ V)
86		relexp0idm 43692	. . . . . . . . . . 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 1193	. . . . . . . . . . . 12 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐽 = 0)
89	88	oveq2d 7365	. . . . . . . . . . 11 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐽) = ((𝐴 × 𝐵)↑_𝑟0))
90		simpl1 1192	. . . . . . . . . . 11 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐾 = 0)
91	89, 90	oveq12d 7367	. . . . . . . . . 10 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = (((𝐴 × 𝐵)↑_𝑟0)↑_𝑟0))
92		simpl3 1194	. . . . . . . . . . 11 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → 𝐼 = 0)
93	92	oveq2d 7365	. . . . . . . . . 10 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → ((𝐴 × 𝐵)↑_𝑟𝐼) = ((𝐴 × 𝐵)↑_𝑟0))
94	87, 91, 93	3eqtr4d 2774	. . . . . . . . 9 ⊢ (((𝐾 = 0 ∧ 𝐽 = 0 ∧ 𝐼 = 0) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅)) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼))
95	94	3exp1 1353	. . . . . . . 8 ⊢ (𝐾 = 0 → (𝐽 = 0 → (𝐼 = 0 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐴 × 𝐵)↑_𝑟𝐽)↑_𝑟𝐾) = ((𝐴 × 𝐵)↑_𝑟𝐼)))))
96	82, 95	jaod 859	. . . . . . 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 1119	. . . . 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 857	. . . 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 217	. . 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 1111	. 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 407	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 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 Vcvv 3436 ∪ cun 3901 ∩ cin 3902 ∅c0 4284 ifcif 4476 class class class wbr 5092 I cid 5513 × cxp 5617 dom cdm 5619 ran crn 5620 ↾ cres 5621 (class class class)co 7349 0cc0 11009 < clt 11149 ℕcn 12128 ℕ₀cn0 12384 ↑_𝑟crelexp 14926

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-er 8625 df-en 8873 df-dom 8874 df-sdom 8875 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-nn 12129 df-n0 12385 df-z 12472 df-uz 12736 df-seq 13909 df-relexp 14927

This theorem is referenced by: (None)

W3C validator