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| Mirrors > Home > MPE Home > Th. List > Mathboxes > esumeq2dv | Structured version Visualization version GIF version | ||
| Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 2-Jan-2017.) | 
| Ref | Expression | 
|---|---|
| esumeq2dv.1 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 = 𝐶) | 
| Ref | Expression | 
|---|---|
| esumeq2dv | ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nfv 1914 | . 2 ⊢ Ⅎ𝑘𝜑 | |
| 2 | esumeq2dv.1 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 = 𝐶) | |
| 3 | 2 | ralrimiva 3146 | . 2 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 = 𝐶) | 
| 4 | 1, 3 | esumeq2d 34038 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Σ*cesum 34028 | 
| 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-10 2141 ax-12 2177 ax-ext 2708 | 
| 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-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-iota 6514 df-fv 6569 df-ov 7434 df-esum 34029 | 
| This theorem is referenced by: esumeq2sdv 34040 esumle 34059 esummulc1 34082 esummulc2 34083 esumdivc 34084 esumsup 34090 measinb 34222 measres 34223 measdivcst 34225 measdivcstALTV 34226 cntmeas 34227 ddemeas 34237 omsval 34295 totprobd 34428 | 
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