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Mirrors > Home > MPE Home > Th. List > sumfc | Structured version Visualization version GIF version |
Description: A lemma to facilitate conversions from the function form to the class-variable form of a sum. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Mario Carneiro, 23-Apr-2014.) |
Ref | Expression |
---|---|
sumfc | ⊢ Σ𝑗 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑗) = Σ𝑘 ∈ 𝐴 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . . . 4 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵) | |
2 | 1 | fvmpt2i 6199 | . . 3 ⊢ (𝑘 ∈ 𝐴 → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) = ( I ‘𝐵)) |
3 | 2 | sumeq2i 14277 | . 2 ⊢ Σ𝑘 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) = Σ𝑘 ∈ 𝐴 ( I ‘𝐵) |
4 | nffvmpt1 6111 | . . 3 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑗) | |
5 | nfcv 2751 | . . 3 ⊢ Ⅎ𝑗((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) | |
6 | fveq2 6103 | . . 3 ⊢ (𝑗 = 𝑘 → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑗) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘)) | |
7 | 4, 5, 6 | cbvsumi 14275 | . 2 ⊢ Σ𝑗 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑗) = Σ𝑘 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) |
8 | sum2id 14286 | . 2 ⊢ Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 ( I ‘𝐵) | |
9 | 3, 7, 8 | 3eqtr4i 2642 | 1 ⊢ Σ𝑗 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑗) = Σ𝑘 ∈ 𝐴 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ↦ cmpt 4643 I cid 4948 ‘cfv 5804 Σcsu 14264 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-seq 12664 df-sum 14265 |
This theorem is referenced by: fsumf1o 14301 sumss 14302 fsumss 14303 fsumcl2lem 14309 fsumadd 14317 isumclim3 14332 isummulc2 14335 fsummulc2 14358 fsumrelem 14380 isumshft 14410 fprodefsum 14664 gsumfsum 19632 |
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