![]() |
Mathbox for Gino Giotto |
< Previous
Next >
Nearby theorems |
|
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 15718 | . 2 ⊢ Σ𝑗 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐴 𝐷 |
3 | cbvsumvw2.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
4 | 3 | sumeq1i 15719 | . 2 ⊢ Σ𝑘 ∈ 𝐴 𝐷 = Σ𝑘 ∈ 𝐵 𝐷 |
5 | 2, 4 | eqtri 2761 | 1 ⊢ Σ𝑗 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐷 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1535 Σcsu 15708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-ext 2704 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-sb 2061 df-clab 2711 df-cleq 2725 df-clel 2812 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-br 5150 df-opab 5212 df-mpt 5233 df-xp 5689 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6317 df-iota 6510 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-ov 7428 df-oprab 7429 df-mpo 7430 df-frecs 8299 df-wrecs 8330 df-recs 8404 df-rdg 8443 df-seq 14029 df-sum 15709 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |