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Mirrors > Home > MPE Home > Th. List > prdsplusgval | Structured version Visualization version GIF version |
Description: Value of a componentwise sum in a structure product. (Contributed by Stefan O'Rear, 10-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-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) |
Ref | Expression |
---|---|
prdsplusgval | ⊢ (𝜑 → (𝐹 + 𝐺) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(+g‘(𝑅‘𝑥))(𝐺‘𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prdsbasmpt.y | . . 3 ⊢ 𝑌 = (𝑆Xs𝑅) | |
2 | prdsbasmpt.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
3 | prdsbasmpt.r | . . . 4 ⊢ (𝜑 → 𝑅 Fn 𝐼) | |
4 | prdsbasmpt.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
5 | fnex 6753 | . . . 4 ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑊) → 𝑅 ∈ V) | |
6 | 3, 4, 5 | syl2anc 579 | . . 3 ⊢ (𝜑 → 𝑅 ∈ V) |
7 | prdsbasmpt.b | . . 3 ⊢ 𝐵 = (Base‘𝑌) | |
8 | fndm 6235 | . . . 4 ⊢ (𝑅 Fn 𝐼 → dom 𝑅 = 𝐼) | |
9 | 3, 8 | syl 17 | . . 3 ⊢ (𝜑 → dom 𝑅 = 𝐼) |
10 | prdsplusgval.p | . . 3 ⊢ + = (+g‘𝑌) | |
11 | 1, 2, 6, 7, 9, 10 | prdsplusg 16504 | . 2 ⊢ (𝜑 → + = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑦‘𝑥)(+g‘(𝑅‘𝑥))(𝑧‘𝑥))))) |
12 | fveq1 6445 | . . . . 5 ⊢ (𝑦 = 𝐹 → (𝑦‘𝑥) = (𝐹‘𝑥)) | |
13 | fveq1 6445 | . . . . 5 ⊢ (𝑧 = 𝐺 → (𝑧‘𝑥) = (𝐺‘𝑥)) | |
14 | 12, 13 | oveqan12d 6941 | . . . 4 ⊢ ((𝑦 = 𝐹 ∧ 𝑧 = 𝐺) → ((𝑦‘𝑥)(+g‘(𝑅‘𝑥))(𝑧‘𝑥)) = ((𝐹‘𝑥)(+g‘(𝑅‘𝑥))(𝐺‘𝑥))) |
15 | 14 | adantl 475 | . . 3 ⊢ ((𝜑 ∧ (𝑦 = 𝐹 ∧ 𝑧 = 𝐺)) → ((𝑦‘𝑥)(+g‘(𝑅‘𝑥))(𝑧‘𝑥)) = ((𝐹‘𝑥)(+g‘(𝑅‘𝑥))(𝐺‘𝑥))) |
16 | 15 | mpteq2dv 4980 | . 2 ⊢ ((𝜑 ∧ (𝑦 = 𝐹 ∧ 𝑧 = 𝐺)) → (𝑥 ∈ 𝐼 ↦ ((𝑦‘𝑥)(+g‘(𝑅‘𝑥))(𝑧‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(+g‘(𝑅‘𝑥))(𝐺‘𝑥)))) |
17 | prdsplusgval.f | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
18 | prdsplusgval.g | . 2 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
19 | 4 | mptexd 6759 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(+g‘(𝑅‘𝑥))(𝐺‘𝑥))) ∈ V) |
20 | 11, 16, 17, 18, 19 | ovmpt2d 7065 | 1 ⊢ (𝜑 → (𝐹 + 𝐺) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(+g‘(𝑅‘𝑥))(𝐺‘𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 Vcvv 3397 ↦ cmpt 4965 dom cdm 5355 Fn wfn 6130 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 +gcplusg 16338 Xscprds 16492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-sup 8636 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-hom 16362 df-cco 16363 df-prds 16494 |
This theorem is referenced by: prdsplusgfval 16520 pwsplusgval 16536 xpsadd 16622 prdsplusgcl 17707 prdsidlem 17708 prdsmndd 17709 prdsinvlem 17911 prdscmnd 18650 prdsringd 18999 prdslmodd 19364 prdstmdd 22335 |
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