Theorem cbvsumdavw2 36234

Description: Change bound variable and the set of integers in a sum. Deduction form. (Contributed by GG, 14-Aug-2025.)

Hypotheses

Ref	Expression
cbvsumdavw2.1	⊢ (𝜑 → 𝐴 = 𝐵)
cbvsumdavw2.2	⊢ ((𝜑 ∧ 𝑗 = 𝑘) → 𝐶 = 𝐷)

Assertion

Ref	Expression
cbvsumdavw2	⊢ (𝜑 → Σ𝑗 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐷)

Distinct variable groups:   𝜑,𝑗,𝑘   𝐶,𝑘   𝐷,𝑗

Allowed substitution hints:   𝐴(𝑗,𝑘)   𝐵(𝑗,𝑘)   𝐶(𝑗)   𝐷(𝑘)

Proof of Theorem cbvsumdavw2

Dummy variables 𝑥 𝑚 𝑛 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvsumdavw2.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 = 𝐵)
2	1	sseq1d 3988	. . . . . 6 ⊢ (𝜑 → (𝐴 ⊆ (ℤ≥‘𝑚) ↔ 𝐵 ⊆ (ℤ≥‘𝑚)))
3	1	eleq2d 2819	. . . . . . . . . 10 ⊢ (𝜑 → (𝑛 ∈ 𝐴 ↔ 𝑛 ∈ 𝐵))
4		cbvsumdavw2.2	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 = 𝑘) → 𝐶 = 𝐷)
5	4	cbvcsbdavw 36198	. . . . . . . . . 10 ⊢ (𝜑 → ⦋𝑛 / 𝑗⦌𝐶 = ⦋𝑛 / 𝑘⦌𝐷)
6	3, 5	ifbieq1d 4523	. . . . . . . . 9 ⊢ (𝜑 → if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑗⦌𝐶, 0) = if(𝑛 ∈ 𝐵, ⦋𝑛 / 𝑘⦌𝐷, 0))
7	6	mpteq2dv 5212	. . . . . . . 8 ⊢ (𝜑 → (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑗⦌𝐶, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐵, ⦋𝑛 / 𝑘⦌𝐷, 0)))
8	7	seqeq3d 14016	. . . . . . 7 ⊢ (𝜑 → seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑗⦌𝐶, 0))) = seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐵, ⦋𝑛 / 𝑘⦌𝐷, 0))))
9	8	breq1d 5126	. . . . . 6 ⊢ (𝜑 → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑗⦌𝐶, 0))) ⇝ 𝑥 ↔ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐵, ⦋𝑛 / 𝑘⦌𝐷, 0))) ⇝ 𝑥))
10	2, 9	anbi12d 632	. . . . 5 ⊢ (𝜑 → ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑗⦌𝐶, 0))) ⇝ 𝑥) ↔ (𝐵 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐵, ⦋𝑛 / 𝑘⦌𝐷, 0))) ⇝ 𝑥)))
11	10	rexbidv 3162	. . . 4 ⊢ (𝜑 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑗⦌𝐶, 0))) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐵 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐵, ⦋𝑛 / 𝑘⦌𝐷, 0))) ⇝ 𝑥)))
12	1	f1oeq3d 6811	. . . . . . 7 ⊢ (𝜑 → (𝑓:(1...𝑚)–1-1-onto→𝐴 ↔ 𝑓:(1...𝑚)–1-1-onto→𝐵))
13	4	cbvcsbdavw 36198	. . . . . . . . . . 11 ⊢ (𝜑 → ⦋(𝑓‘𝑛) / 𝑗⦌𝐶 = ⦋(𝑓‘𝑛) / 𝑘⦌𝐷)
14	13	mpteq2dv 5212	. . . . . . . . . 10 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐶) = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐷))
15	14	seqeq3d 14016	. . . . . . . . 9 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐶)) = seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐷)))
16	15	fveq1d 6874	. . . . . . . 8 ⊢ (𝜑 → (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐶))‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐷))‘𝑚))
17	16	eqeq2d 2745	. . . . . . 7 ⊢ (𝜑 → (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐶))‘𝑚) ↔ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐷))‘𝑚)))
18	12, 17	anbi12d 632	. . . . . 6 ⊢ (𝜑 → ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐶))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto→𝐵 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐷))‘𝑚))))
19	18	exbidv 1920	. . . . 5 ⊢ (𝜑 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐶))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐵 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐷))‘𝑚))))
20	19	rexbidv 3162	. . . 4 ⊢ (𝜑 → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐶))‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐵 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐷))‘𝑚))))
21	11, 20	orbi12d 918	. . 3 ⊢ (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑗⦌𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐶))‘𝑚))) ↔ (∃𝑚 ∈ ℤ (𝐵 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐵, ⦋𝑛 / 𝑘⦌𝐷, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐵 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐷))‘𝑚)))))
22	21	iotabidv 6511	. 2 ⊢ (𝜑 → (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑗⦌𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐶))‘𝑚)))) = (℩𝑥(∃𝑚 ∈ ℤ (𝐵 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐵, ⦋𝑛 / 𝑘⦌𝐷, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐵 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐷))‘𝑚)))))
23		df-sum 15690	. 2 ⊢ Σ𝑗 ∈ 𝐴 𝐶 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑗⦌𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐶))‘𝑚))))
24		df-sum 15690	. 2 ⊢ Σ𝑘 ∈ 𝐵 𝐷 = (℩𝑥(∃𝑚 ∈ ℤ (𝐵 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐵, ⦋𝑛 / 𝑘⦌𝐷, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐵 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐷))‘𝑚))))
25	22, 23, 24	3eqtr4g 2794	1 ⊢ (𝜑 → Σ𝑗 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐷)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1539  ∃wex 1778   ∈ wcel 2107  ∃wrex 3059  ⦋csb 3872   ⊆ wss 3924  ifcif 4498   class class class wbr 5116   ↦ cmpt 5198  ℩cio 6478  –1-1-onto→wf1o 6526  ‘cfv 6527  (class class class)co 7399  0cc0 11121  1c1 11122   + caddc 11124  ℕcn 12232  ℤcz 12580  ℤ_≥cuz 12844  ...cfz 13513  seqcseq 14008   ⇝ cli 15487  Σcsu 15689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4881  df-br 5117  df-opab 5179  df-mpt 5199  df-xp 5657  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6287  df-iota 6480  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7402  df-oprab 7403  df-mpo 7404  df-frecs 8274  df-wrecs 8305  df-recs 8379  df-rdg 8418  df-seq 14009  df-sum 15690

This theorem is referenced by: (None)