Theorem itgeq1

Description: Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014)

	Ref	Expression
Assertion	itgeq1	$$⊢A=B\to {\int }_{A}Cdx={\int }_{B}Cdx$$

Proof

Step	Hyp	Ref	Expression
1		eleq2	$$⊢A=B\to \left(x\in A\leftrightarrow x\in B\right)$$
2	1	anbi1d	$$⊢A=B\to \left(\left(x\in A\wedge 0\le y\right)\leftrightarrow \left(x\in B\wedge 0\le y\right)\right)$$
3	2	ifbid	$$⊢A=B\to if\left(\left(x\in A\wedge 0\le y\right),y,0\right)=if\left(\left(x\in B\wedge 0\le y\right),y,0\right)$$
4	3	csbeq2dv	$$⊢A=B\to ⦋\Re \left(\frac{C}{{\mathrm{i}}^k}\right)/y⦌if\left(\left(x\in A\wedge 0\le y\right),y,0\right)=⦋\Re \left(\frac{C}{{\mathrm{i}}^k}\right)/y⦌if\left(\left(x\in B\wedge 0\le y\right),y,0\right)$$
5	4	mpteq2dv	$$⊢A=B\to \left(x\in ℝ⟼⦋\Re \left(\frac{C}{{\mathrm{i}}^k}\right)/y⦌if\left(\left(x\in A\wedge 0\le y\right),y,0\right)\right)=\left(x\in ℝ⟼⦋\Re \left(\frac{C}{{\mathrm{i}}^k}\right)/y⦌if\left(\left(x\in B\wedge 0\le y\right),y,0\right)\right)$$
6	5	fveq2d	$$⊢A=B\to {\int }^2\left(\left(x\in ℝ⟼⦋\Re \left(\frac{C}{{\mathrm{i}}^k}\right)/y⦌if\left(\left(x\in A\wedge 0\le y\right),y,0\right)\right)\right)={\int }^2\left(\left(x\in ℝ⟼⦋\Re \left(\frac{C}{{\mathrm{i}}^k}\right)/y⦌if\left(\left(x\in B\wedge 0\le y\right),y,0\right)\right)\right)$$
7	6	oveq2d	$$⊢A=B\to {\mathrm{i}}^k{\int }^2\left(\left(x\in ℝ⟼⦋\Re \left(\frac{C}{{\mathrm{i}}^k}\right)/y⦌if\left(\left(x\in A\wedge 0\le y\right),y,0\right)\right)\right)={\mathrm{i}}^k{\int }^2\left(\left(x\in ℝ⟼⦋\Re \left(\frac{C}{{\mathrm{i}}^k}\right)/y⦌if\left(\left(x\in B\wedge 0\le y\right),y,0\right)\right)\right)$$
8	7	sumeq2sdv	$$⊢A=B\to \sum _{k=0}^3{\mathrm{i}}^k{\int }^2\left(\left(x\in ℝ⟼⦋\Re \left(\frac{C}{{\mathrm{i}}^k}\right)/y⦌if\left(\left(x\in A\wedge 0\le y\right),y,0\right)\right)\right)=\sum _{k=0}^3{\mathrm{i}}^k{\int }^2\left(\left(x\in ℝ⟼⦋\Re \left(\frac{C}{{\mathrm{i}}^k}\right)/y⦌if\left(\left(x\in B\wedge 0\le y\right),y,0\right)\right)\right)$$
9		df-itg	$$⊢{\int }_{A}Cdx=\sum _{k=0}^3{\mathrm{i}}^k{\int }^2\left(\left(x\in ℝ⟼⦋\Re \left(\frac{C}{{\mathrm{i}}^k}\right)/y⦌if\left(\left(x\in A\wedge 0\le y\right),y,0\right)\right)\right)$$
10		df-itg	$$⊢{\int }_{B}Cdx=\sum _{k=0}^3{\mathrm{i}}^k{\int }^2\left(\left(x\in ℝ⟼⦋\Re \left(\frac{C}{{\mathrm{i}}^k}\right)/y⦌if\left(\left(x\in B\wedge 0\le y\right),y,0\right)\right)\right)$$
11	8 9 10	3eqtr4g	$$⊢A=B\to {\int }_{A}Cdx={\int }_{B}Cdx$$