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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cbvsumvw2 | Structured version Visualization version GIF version | ||
| Description: Change bound variable and the set of integers in a sum, using implicit substitution. (Contributed by GG, 1-Sep-2025.) |
| Ref | Expression |
|---|---|
| cbvsumvw2.1 | ⊢ 𝐴 = 𝐵 |
| cbvsumvw2.2 | ⊢ (𝑗 = 𝑘 → 𝐶 = 𝐷) |
| Ref | Expression |
|---|---|
| cbvsumvw2 | ⊢ Σ𝑗 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐷 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cbvsumvw2.2 | . . 3 ⊢ (𝑗 = 𝑘 → 𝐶 = 𝐷) | |
| 2 | 1 | cbvsumv 15728 | . 2 ⊢ Σ𝑗 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐴 𝐷 |
| 3 | cbvsumvw2.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
| 4 | 3 | sumeq1i 15729 | . 2 ⊢ Σ𝑘 ∈ 𝐴 𝐷 = Σ𝑘 ∈ 𝐵 𝐷 |
| 5 | 2, 4 | eqtri 2764 | 1 ⊢ Σ𝑗 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐷 |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 Σcsu 15718 |
| 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 2007 ax-8 2110 ax-9 2118 ax-ext 2707 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-opab 5204 df-mpt 5224 df-xp 5689 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-iota 6512 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-ov 7432 df-oprab 7433 df-mpo 7434 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-seq 14039 df-sum 15719 |
| This theorem is referenced by: (None) |
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