Theorem itgeq12i

Description: Equality inference for an integral. General version of itgeq1i and itgeq2i . (Contributed by GG, 1-Sep-2025)

	Ref	Expression
Hypotheses	itgeq12i.1	$$⊢A=B$$
	itgeq12i.2	$$⊢C=D$$
Assertion	itgeq12i	$$⊢{\int }_{A}Cdx={\int }_{B}Ddx$$

Proof

Step	Hyp	Ref	Expression
1		itgeq12i.1	$$⊢A=B$$
2		itgeq12i.2	$$⊢C=D$$
3	2	oveq1i	$$⊢\frac{C}{{\mathrm{i}}^k}=\frac{D}{{\mathrm{i}}^k}$$
4	3	fveq2i	$$⊢\Re \left(\frac{C}{{\mathrm{i}}^k}\right)=\Re \left(\frac{D}{{\mathrm{i}}^k}\right)$$
5	1	eleq2i	$$⊢x\in A\leftrightarrow x\in B$$
6	5	anbi1i	$$⊢\left(x\in A\wedge 0\le y\right)\leftrightarrow \left(x\in B\wedge 0\le y\right)$$
7		ifbi	$$⊢\left(\left(x\in A\wedge 0\le y\right)\leftrightarrow \left(x\in B\wedge 0\le y\right)\right)\to if\left(\left(x\in A\wedge 0\le y\right),y,0\right)=if\left(\left(x\in B\wedge 0\le y\right),y,0\right)$$
8	6 7	ax-mp	$$⊢if\left(\left(x\in A\wedge 0\le y\right),y,0\right)=if\left(\left(x\in B\wedge 0\le y\right),y,0\right)$$
9	8	ax-gen	$$⊢\forall yif\left(\left(x\in A\wedge 0\le y\right),y,0\right)=if\left(\left(x\in B\wedge 0\le y\right),y,0\right)$$
10	4 9	pm3.2i	$$⊢\left(\Re \left(\frac{C}{{\mathrm{i}}^k}\right)=\Re \left(\frac{D}{{\mathrm{i}}^k}\right)\wedge \forall yif\left(\left(x\in A\wedge 0\le y\right),y,0\right)=if\left(\left(x\in B\wedge 0\le y\right),y,0\right)\right)$$
11		csbeq2	$$⊢\forall yif\left(\left(x\in A\wedge 0\le y\right),y,0\right)=if\left(\left(x\in B\wedge 0\le y\right),y,0\right)\to ⦋\Re \left(\frac{C}{{\mathrm{i}}^k}\right)/y⦌if\left(\left(x\in A\wedge 0\le y\right),y,0\right)=⦋\Re \left(\frac{C}{{\mathrm{i}}^k}\right)/y⦌if\left(\left(x\in B\wedge 0\le y\right),y,0\right)$$
12		csbeq1	$$⊢\Re \left(\frac{C}{{\mathrm{i}}^k}\right)=\Re \left(\frac{D}{{\mathrm{i}}^k}\right)\to ⦋\Re \left(\frac{C}{{\mathrm{i}}^k}\right)/y⦌if\left(\left(x\in B\wedge 0\le y\right),y,0\right)=⦋\Re \left(\frac{D}{{\mathrm{i}}^k}\right)/y⦌if\left(\left(x\in B\wedge 0\le y\right),y,0\right)$$
13	11 12	sylan9eqr	$$⊢\left(\Re \left(\frac{C}{{\mathrm{i}}^k}\right)=\Re \left(\frac{D}{{\mathrm{i}}^k}\right)\wedge \forall yif\left(\left(x\in A\wedge 0\le y\right),y,0\right)=if\left(\left(x\in B\wedge 0\le y\right),y,0\right)\right)\to ⦋\Re \left(\frac{C}{{\mathrm{i}}^k}\right)/y⦌if\left(\left(x\in A\wedge 0\le y\right),y,0\right)=⦋\Re \left(\frac{D}{{\mathrm{i}}^k}\right)/y⦌if\left(\left(x\in B\wedge 0\le y\right),y,0\right)$$
14	10 13	ax-mp	$$⊢⦋\Re \left(\frac{C}{{\mathrm{i}}^k}\right)/y⦌if\left(\left(x\in A\wedge 0\le y\right),y,0\right)=⦋\Re \left(\frac{D}{{\mathrm{i}}^k}\right)/y⦌if\left(\left(x\in B\wedge 0\le y\right),y,0\right)$$
15	14	mpteq2i	$$⊢\left(x\in ℝ⟼⦋\Re \left(\frac{C}{{\mathrm{i}}^k}\right)/y⦌if\left(\left(x\in A\wedge 0\le y\right),y,0\right)\right)=\left(x\in ℝ⟼⦋\Re \left(\frac{D}{{\mathrm{i}}^k}\right)/y⦌if\left(\left(x\in B\wedge 0\le y\right),y,0\right)\right)$$
16	15	fveq2i	$$⊢{\int }^2\left(\left(x\in ℝ⟼⦋\Re \left(\frac{C}{{\mathrm{i}}^k}\right)/y⦌if\left(\left(x\in A\wedge 0\le y\right),y,0\right)\right)\right)={\int }^2\left(\left(x\in ℝ⟼⦋\Re \left(\frac{D}{{\mathrm{i}}^k}\right)/y⦌if\left(\left(x\in B\wedge 0\le y\right),y,0\right)\right)\right)$$
17	16	oveq2i	$$⊢{\mathrm{i}}^k{\int }^2\left(\left(x\in ℝ⟼⦋\Re \left(\frac{C}{{\mathrm{i}}^k}\right)/y⦌if\left(\left(x\in A\wedge 0\le y\right),y,0\right)\right)\right)={\mathrm{i}}^k{\int }^2\left(\left(x\in ℝ⟼⦋\Re \left(\frac{D}{{\mathrm{i}}^k}\right)/y⦌if\left(\left(x\in B\wedge 0\le y\right),y,0\right)\right)\right)$$
18	17	sumeq2si	$$⊢\sum _{k=0}^3{\mathrm{i}}^k{\int }^2\left(\left(x\in ℝ⟼⦋\Re \left(\frac{C}{{\mathrm{i}}^k}\right)/y⦌if\left(\left(x\in A\wedge 0\le y\right),y,0\right)\right)\right)=\sum _{k=0}^3{\mathrm{i}}^k{\int }^2\left(\left(x\in ℝ⟼⦋\Re \left(\frac{D}{{\mathrm{i}}^k}\right)/y⦌if\left(\left(x\in B\wedge 0\le y\right),y,0\right)\right)\right)$$
19		df-itg	$$⊢{\int }_{A}Cdx=\sum _{k=0}^3{\mathrm{i}}^k{\int }^2\left(\left(x\in ℝ⟼⦋\Re \left(\frac{C}{{\mathrm{i}}^k}\right)/y⦌if\left(\left(x\in A\wedge 0\le y\right),y,0\right)\right)\right)$$
20		df-itg	$$⊢{\int }_{B}Ddx=\sum _{k=0}^3{\mathrm{i}}^k{\int }^2\left(\left(x\in ℝ⟼⦋\Re \left(\frac{D}{{\mathrm{i}}^k}\right)/y⦌if\left(\left(x\in B\wedge 0\le y\right),y,0\right)\right)\right)$$
21	18 19 20	3eqtr4i	$$⊢{\int }_{A}Cdx={\int }_{B}Ddx$$