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Mirrors > Home > MPE Home > Th. List > prdsplusgfval | Structured version Visualization version GIF version |
Description: Value of a structure product sum at a single coordinate. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
Ref | Expression |
---|---|
prdsbasmpt.y | ⊢ 𝑌 = (𝑆Xs𝑅) |
prdsbasmpt.b | ⊢ 𝐵 = (Base‘𝑌) |
prdsbasmpt.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
prdsbasmpt.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
prdsbasmpt.r | ⊢ (𝜑 → 𝑅 Fn 𝐼) |
prdsplusgval.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
prdsplusgval.g | ⊢ (𝜑 → 𝐺 ∈ 𝐵) |
prdsplusgval.p | ⊢ + = (+g‘𝑌) |
prdsplusgfval.j | ⊢ (𝜑 → 𝐽 ∈ 𝐼) |
Ref | Expression |
---|---|
prdsplusgfval | ⊢ (𝜑 → ((𝐹 + 𝐺)‘𝐽) = ((𝐹‘𝐽)(+g‘(𝑅‘𝐽))(𝐺‘𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prdsbasmpt.y | . . . 4 ⊢ 𝑌 = (𝑆Xs𝑅) | |
2 | prdsbasmpt.b | . . . 4 ⊢ 𝐵 = (Base‘𝑌) | |
3 | prdsbasmpt.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
4 | prdsbasmpt.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
5 | prdsbasmpt.r | . . . 4 ⊢ (𝜑 → 𝑅 Fn 𝐼) | |
6 | prdsplusgval.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
7 | prdsplusgval.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
8 | prdsplusgval.p | . . . 4 ⊢ + = (+g‘𝑌) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | prdsplusgval 16842 | . . 3 ⊢ (𝜑 → (𝐹 + 𝐺) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(+g‘(𝑅‘𝑥))(𝐺‘𝑥)))) |
10 | 9 | fveq1d 6670 | . 2 ⊢ (𝜑 → ((𝐹 + 𝐺)‘𝐽) = ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(+g‘(𝑅‘𝑥))(𝐺‘𝑥)))‘𝐽)) |
11 | prdsplusgfval.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝐼) | |
12 | 2fveq3 6673 | . . . . 5 ⊢ (𝑥 = 𝐽 → (+g‘(𝑅‘𝑥)) = (+g‘(𝑅‘𝐽))) | |
13 | fveq2 6668 | . . . . 5 ⊢ (𝑥 = 𝐽 → (𝐹‘𝑥) = (𝐹‘𝐽)) | |
14 | fveq2 6668 | . . . . 5 ⊢ (𝑥 = 𝐽 → (𝐺‘𝑥) = (𝐺‘𝐽)) | |
15 | 12, 13, 14 | oveq123d 7185 | . . . 4 ⊢ (𝑥 = 𝐽 → ((𝐹‘𝑥)(+g‘(𝑅‘𝑥))(𝐺‘𝑥)) = ((𝐹‘𝐽)(+g‘(𝑅‘𝐽))(𝐺‘𝐽))) |
16 | eqid 2738 | . . . 4 ⊢ (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(+g‘(𝑅‘𝑥))(𝐺‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(+g‘(𝑅‘𝑥))(𝐺‘𝑥))) | |
17 | ovex 7197 | . . . 4 ⊢ ((𝐹‘𝐽)(+g‘(𝑅‘𝐽))(𝐺‘𝐽)) ∈ V | |
18 | 15, 16, 17 | fvmpt 6769 | . . 3 ⊢ (𝐽 ∈ 𝐼 → ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(+g‘(𝑅‘𝑥))(𝐺‘𝑥)))‘𝐽) = ((𝐹‘𝐽)(+g‘(𝑅‘𝐽))(𝐺‘𝐽))) |
19 | 11, 18 | syl 17 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(+g‘(𝑅‘𝑥))(𝐺‘𝑥)))‘𝐽) = ((𝐹‘𝐽)(+g‘(𝑅‘𝐽))(𝐺‘𝐽))) |
20 | 10, 19 | eqtrd 2773 | 1 ⊢ (𝜑 → ((𝐹 + 𝐺)‘𝐽) = ((𝐹‘𝐽)(+g‘(𝑅‘𝐽))(𝐺‘𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 ↦ cmpt 5107 Fn wfn 6328 ‘cfv 6333 (class class class)co 7164 Basecbs 16579 +gcplusg 16661 Xscprds 16815 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-er 8313 df-map 8432 df-ixp 8501 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-sup 8972 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-7 11777 df-8 11778 df-9 11779 df-n0 11970 df-z 12056 df-dec 12173 df-uz 12318 df-fz 12975 df-struct 16581 df-ndx 16582 df-slot 16583 df-base 16585 df-plusg 16674 df-mulr 16675 df-sca 16677 df-vsca 16678 df-ip 16679 df-tset 16680 df-ple 16681 df-ds 16683 df-hom 16685 df-cco 16686 df-prds 16817 |
This theorem is referenced by: prdsmndd 18053 prdspjmhm 18102 prdsringd 19477 prdslmodd 19853 dsmmacl 20550 |
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