![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 17360 | . . 3 ⊢ (𝜑 → (𝐹 + 𝐺) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(+g‘(𝑅‘𝑥))(𝐺‘𝑥)))) |
10 | 9 | fveq1d 6845 | . 2 ⊢ (𝜑 → ((𝐹 + 𝐺)‘𝐽) = ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(+g‘(𝑅‘𝑥))(𝐺‘𝑥)))‘𝐽)) |
11 | prdsplusgfval.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝐼) | |
12 | 2fveq3 6848 | . . . . 5 ⊢ (𝑥 = 𝐽 → (+g‘(𝑅‘𝑥)) = (+g‘(𝑅‘𝐽))) | |
13 | fveq2 6843 | . . . . 5 ⊢ (𝑥 = 𝐽 → (𝐹‘𝑥) = (𝐹‘𝐽)) | |
14 | fveq2 6843 | . . . . 5 ⊢ (𝑥 = 𝐽 → (𝐺‘𝑥) = (𝐺‘𝐽)) | |
15 | 12, 13, 14 | oveq123d 7379 | . . . 4 ⊢ (𝑥 = 𝐽 → ((𝐹‘𝑥)(+g‘(𝑅‘𝑥))(𝐺‘𝑥)) = ((𝐹‘𝐽)(+g‘(𝑅‘𝐽))(𝐺‘𝐽))) |
16 | eqid 2733 | . . . 4 ⊢ (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(+g‘(𝑅‘𝑥))(𝐺‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(+g‘(𝑅‘𝑥))(𝐺‘𝑥))) | |
17 | ovex 7391 | . . . 4 ⊢ ((𝐹‘𝐽)(+g‘(𝑅‘𝐽))(𝐺‘𝐽)) ∈ V | |
18 | 15, 16, 17 | fvmpt 6949 | . . 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 2107 ↦ cmpt 5189 Fn wfn 6492 ‘cfv 6497 (class class class)co 7358 Basecbs 17088 +gcplusg 17138 Xscprds 17332 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-fz 13431 df-struct 17024 df-slot 17059 df-ndx 17071 df-base 17089 df-plusg 17151 df-mulr 17152 df-sca 17154 df-vsca 17155 df-ip 17156 df-tset 17157 df-ple 17158 df-ds 17160 df-hom 17162 df-cco 17163 df-prds 17334 |
This theorem is referenced by: prdsmndd 18594 prdspjmhm 18644 prdsringd 20041 prdslmodd 20445 dsmmacl 21163 |
Copyright terms: Public domain | W3C validator |