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Related theorems GIF version |
| Description: Separate the nested sum of the product A(j) · B(m). |
| Ref | Expression |
|---|---|
| fsum2mul | ⊢ (((K ∈ (ℤ≥ ‘J) ⋀ ∀j ∈ (J...K)A ∈ ℂ) ⋀ (N ∈ (ℤ≥ ‘M) ⋀ ∀m ∈ (M...N)B ∈ ℂ)) → Σj ∈ (J...K)Σm ∈ (M...N)(A · B) = (Σj ∈ (J...K)A · Σm ∈ (M...N)B)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fsummulc2 7034 | . . . . 5 ⊢ ((K ∈ (ℤ≥ ‘J) ⋀ Σm ∈ (M...N)B ∈ ℂ ⋀ ∀j ∈ (J...K)A ∈ ℂ) → (Σj ∈ (J...K)A · Σm ∈ (M...N)B) = Σj ∈ (J...K)(A · Σm ∈ (M...N)B)) | |
| 2 | 1 | 3expa 835 | . . . 4 ⊢ (((K ∈ (ℤ≥ ‘J) ⋀ Σm ∈ (M...N)B ∈ ℂ) ⋀ ∀j ∈ (J...K)A ∈ ℂ) → (Σj ∈ (J...K)A · Σm ∈ (M...N)B) = Σj ∈ (J...K)(A · Σm ∈ (M...N)B)) |
| 3 | 2 | an1rs 491 | . . 3 ⊢ (((K ∈ (ℤ≥ ‘J) ⋀ ∀j ∈ (J...K)A ∈ ℂ) ⋀ Σm ∈ (M...N)B ∈ ℂ) → (Σj ∈ (J...K)A · Σm ∈ (M...N)B) = Σj ∈ (J...K)(A · Σm ∈ (M...N)B)) |
| 4 | fsumclt 7015 | . . 3 ⊢ ((N ∈ (ℤ≥ ‘M) ⋀ ∀m ∈ (M...N)B ∈ ℂ) → Σm ∈ (M...N)B ∈ ℂ) | |
| 5 | 3, 4 | sylan2 453 | . 2 ⊢ (((K ∈ (ℤ≥ ‘J) ⋀ ∀j ∈ (J...K)A ∈ ℂ) ⋀ (N ∈ (ℤ≥ ‘M) ⋀ ∀m ∈ (M...N)B ∈ ℂ)) → (Σj ∈ (J...K)A · Σm ∈ (M...N)B) = Σj ∈ (J...K)(A · Σm ∈ (M...N)B)) |
| 6 | fsummulc1 7033 | . . . . . . . . 9 ⊢ ((N ∈ (ℤ≥ ‘M) ⋀ A ∈ ℂ ⋀ ∀m ∈ (M...N)B ∈ ℂ) → (A · Σm ∈ (M...N)B) = Σm ∈ (M...N)(A · B)) | |
| 7 | 6 | 3expa 835 | . . . . . . . 8 ⊢ (((N ∈ (ℤ≥ ‘M) ⋀ A ∈ ℂ) ⋀ ∀m ∈ (M...N)B ∈ ℂ) → (A · Σm ∈ (M...N)B) = Σm ∈ (M...N)(A · B)) |
| 8 | 7 | an1rs 491 | . . . . . . 7 ⊢ (((N ∈ (ℤ≥ ‘M) ⋀ ∀m ∈ (M...N)B ∈ ℂ) ⋀ A ∈ ℂ) → (A · Σm ∈ (M...N)B) = Σm ∈ (M...N)(A · B)) |
| 9 | 8 | ex 373 | . . . . . 6 ⊢ ((N ∈ (ℤ≥ ‘M) ⋀ ∀m ∈ (M...N)B ∈ ℂ) → (A ∈ ℂ → (A · Σm ∈ (M...N)B) = Σm ∈ (M...N)(A · B))) |
| 10 | 9 | r19.20sdv 1713 | . . . . 5 ⊢ ((N ∈ (ℤ≥ ‘M) ⋀ ∀m ∈ (M...N)B ∈ ℂ) → (∀j ∈ (J...K)A ∈ ℂ → ∀j ∈ (J...K)(A · Σm ∈ (M...N)B) = Σm ∈ (M...N)(A · B))) |
| 11 | 10 | impcom 351 | . . . 4 ⊢ ((∀j ∈ (J...K)A ∈ ℂ ⋀ (N ∈ (ℤ≥ ‘M) ⋀ ∀m ∈ (M...N)B ∈ ℂ)) → ∀j ∈ (J...K)(A · Σm ∈ (M...N)B) = Σm ∈ (M...N)(A · B)) |
| 12 | 11 | sumeq2d 6991 | . . 3 ⊢ ((∀j ∈ (J...K)A ∈ ℂ ⋀ (N ∈ (ℤ≥ ‘M) ⋀ ∀m ∈ (M...N)B ∈ ℂ)) → Σj ∈ (J...K)(A · Σm ∈ (M...N)B) = Σj ∈ (J...K)Σm ∈ (M...N)(A · B)) |
| 13 | 12 | adantll 394 | . 2 ⊢ (((K ∈ (ℤ≥ ‘J) ⋀ ∀j ∈ (J...K)A ∈ ℂ) ⋀ (N ∈ (ℤ≥ ‘M) ⋀ ∀m ∈ (M...N)B ∈ ℂ)) → Σj ∈ (J...K)(A · Σm ∈ (M...N)B) = Σj ∈ (J...K)Σm ∈ (M...N)(A · B)) |
| 14 | 5, 13 | eqtr2d 1511 | 1 ⊢ (((K ∈ (ℤ≥ ‘J) ⋀ ∀j ∈ (J...K)A ∈ ℂ) ⋀ (N ∈ (ℤ≥ ‘M) ⋀ ∀m ∈ (M...N)B ∈ ℂ)) → Σj ∈ (J...K)Σm ∈ (M...N)(A · B) = (Σj ∈ (J...K)A · Σm ∈ (M...N)B)) |
| Colors of variables: wff set class |
| Syntax hints: → wi 3 ⋀ wa 223 = wceq 958 ∈ wcel 960 ∀wral 1648 ‘cfv 3188 (class class class)co 3969 ℂcc 5244 · cmul 5251 ℤ≥cuz 6418 ...cfz 6468 Σcsu 6979 |
| This theorem is referenced by: fsumabs2mul 7044 efaddlem6 7343 |
| This theorem was proved from axioms: ax-1 4 ax-2 5 ax-3 6 ax-mp 7 ax-7 964 ax-gen 965 ax-8 966 ax-9 967 ax-10 968 ax-11 969 ax-12 970 ax-13 971 ax-14 972 ax-17 973 ax-4 975 ax-5o 977 ax-6o 980 ax-9o 1125 ax-10o 1142 ax-16 1212 ax-11o 1220 ax-ext 1462 ax-rep 2698 ax-sep 2708 ax-nul 2715 ax-pow 2748 ax-pr 2785 ax-un 2872 ax-inf2 4634 |
| This theorem depends on definitions: df-bi 147 df-or 224 df-an 225 df-3or 778 df-3an 779 df-ex 983 df-sb 1174 df-eu 1384 df-mo 1385 df-clab 1467 df-cleq 1472 df-clel 1475 df-ne 1590 df-nel 1591 df-ral 1652 df-rex 1653 df-reu 1654 df-rab 1655 df-v 1815 df-sbc 1945 df-csb 2005 df-dif 2052 df-un 2053 df-in 2054 df-ss 2056 df-pss 2058 df-nul 2284 df-if 2366 df-pw 2406 df-sn 2416 df-pr 2417 df-tp 2419 df-op 2420 df-uni 2508 df-int 2538 df-iun 2572 df-br 2625 df-opab 2672 df-tr 2686 df-eprel 2838 df-id 2841 df-po 2846 df-so 2856 df-fr 2923 df-we 2940 df-ord 2957 df-on 2958 df-lim 2959 df-suc 2960 df-om 3138 df-xp 3190 df-rel 3191 df-cnv 3192 df-co 3193 df-dm 3194 df-rn 3195 df-res 3196 df-ima 3197 df-fun 3198 df-fn 3199 df-f 3200 df-f1 3201 df-fo 3202 df-f1o 3203 df-fv 3204 df-rdg 3938 df-opr 3971 df-oprab 3972 df-1st 4085 df-2nd 4086 df-1o 4139 df-oadd 4141 df-omul 4142 df-er 4267 df-ec 4269 df-qs 4272 df-en 4374 df-dom 4375 df-sdom 4376 df-ni 5012 df-pli 5013 df-mi 5014 df-lti 5015 df-plpq 5047 df-mpq 5048 df-enq 5049 df-nq 5050 df-plq 5051 df-mq 5052 df-rq 5053 df-ltq 5054 df-1q 5055 df-np 5098 df-1p 5099 df-plp 5100 df-mp 5101 df-ltp 5102 df-plpr 5176 df-mpr 5177 df-enr 5178 df-nr 5179 df-plr 5180 df-mr 5181 df-ltr 5182 df-0r 5183 df-1r 5184 df-m1r 5185 df-c 5252 df-0 5253 df-1 5254 df-i 5255 df-r 5256 df-plus 5257 df-mul 5258 df-lt 5259 df-sub 5368 df-neg 5370 df-pnf 5499 df-mnf 5500 df-xr 5501 df-ltxr 5502 df-le 5503 df-n 5927 df-n0 6102 df-z 6138 df-seq1 6309 df-shft 6342 df-uz 6419 df-fz 6469 df-seqz 6534 df-sum 6980 |