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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumeq2d | Structured version Visualization version GIF version |
Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 21-Sep-2016.) |
Ref | Expression |
---|---|
esumeq2d.0 | ⊢ Ⅎ𝑘𝜑 |
esumeq2d.1 | ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 = 𝐶) |
Ref | Expression |
---|---|
esumeq2d | ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | esumeq2d.0 | . 2 ⊢ Ⅎ𝑘𝜑 | |
2 | eqidd 2736 | . 2 ⊢ (𝜑 → 𝐴 = 𝐴) | |
3 | esumeq2d.1 | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 = 𝐶) | |
4 | 3 | r19.21bi 3249 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 = 𝐶) |
5 | 1, 2, 4 | esumeq12dvaf 34012 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 Ⅎwnf 1780 ∀wral 3059 Σ*cesum 34008 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-iota 6516 df-fv 6571 df-ov 7434 df-esum 34009 |
This theorem is referenced by: esumeq2dv 34019 esumpad 34036 esumlef 34043 esumrnmpt2 34049 voliune 34210 omssubadd 34282 carsggect 34300 omsmeas 34305 dstrvprob 34453 |
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