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Mirrors > Home > MPE Home > Th. List > pwsplusgval | Structured version Visualization version GIF version |
Description: Value of addition in a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.) |
Ref | Expression |
---|---|
pwsplusgval.y | ⊢ 𝑌 = (𝑅 ↑s 𝐼) |
pwsplusgval.b | ⊢ 𝐵 = (Base‘𝑌) |
pwsplusgval.r | ⊢ (𝜑 → 𝑅 ∈ 𝑉) |
pwsplusgval.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
pwsplusgval.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
pwsplusgval.g | ⊢ (𝜑 → 𝐺 ∈ 𝐵) |
pwsplusgval.a | ⊢ + = (+g‘𝑅) |
pwsplusgval.p | ⊢ ✚ = (+g‘𝑌) |
Ref | Expression |
---|---|
pwsplusgval | ⊢ (𝜑 → (𝐹 ✚ 𝐺) = (𝐹 ∘f + 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . . 4 ⊢ ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) = ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) | |
2 | eqid 2818 | . . . 4 ⊢ (Base‘((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) = (Base‘((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) | |
3 | fvexd 6678 | . . . 4 ⊢ (𝜑 → (Scalar‘𝑅) ∈ V) | |
4 | pwsplusgval.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
5 | pwsplusgval.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝑉) | |
6 | fnconstg 6560 | . . . . 5 ⊢ (𝑅 ∈ 𝑉 → (𝐼 × {𝑅}) Fn 𝐼) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐼 × {𝑅}) Fn 𝐼) |
8 | pwsplusgval.f | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
9 | pwsplusgval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑌) | |
10 | pwsplusgval.y | . . . . . . . . 9 ⊢ 𝑌 = (𝑅 ↑s 𝐼) | |
11 | eqid 2818 | . . . . . . . . 9 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅) | |
12 | 10, 11 | pwsval 16747 | . . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑌 = ((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) |
13 | 5, 4, 12 | syl2anc 584 | . . . . . . 7 ⊢ (𝜑 → 𝑌 = ((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) |
14 | 13 | fveq2d 6667 | . . . . . 6 ⊢ (𝜑 → (Base‘𝑌) = (Base‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
15 | 9, 14 | syl5eq 2865 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
16 | 8, 15 | eleqtrd 2912 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (Base‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
17 | pwsplusgval.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
18 | 17, 15 | eleqtrd 2912 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (Base‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
19 | eqid 2818 | . . . 4 ⊢ (+g‘((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) = (+g‘((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) | |
20 | 1, 2, 3, 4, 7, 16, 18, 19 | prdsplusgval 16734 | . . 3 ⊢ (𝜑 → (𝐹(+g‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))𝐺) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(+g‘((𝐼 × {𝑅})‘𝑥))(𝐺‘𝑥)))) |
21 | fvconst2g 6956 | . . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ 𝐼) → ((𝐼 × {𝑅})‘𝑥) = 𝑅) | |
22 | 5, 21 | sylan 580 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐼 × {𝑅})‘𝑥) = 𝑅) |
23 | 22 | fveq2d 6667 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (+g‘((𝐼 × {𝑅})‘𝑥)) = (+g‘𝑅)) |
24 | pwsplusgval.a | . . . . . 6 ⊢ + = (+g‘𝑅) | |
25 | 23, 24 | syl6eqr 2871 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (+g‘((𝐼 × {𝑅})‘𝑥)) = + ) |
26 | 25 | oveqd 7162 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥)(+g‘((𝐼 × {𝑅})‘𝑥))(𝐺‘𝑥)) = ((𝐹‘𝑥) + (𝐺‘𝑥))) |
27 | 26 | mpteq2dva 5152 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(+g‘((𝐼 × {𝑅})‘𝑥))(𝐺‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥) + (𝐺‘𝑥)))) |
28 | 20, 27 | eqtrd 2853 | . 2 ⊢ (𝜑 → (𝐹(+g‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))𝐺) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥) + (𝐺‘𝑥)))) |
29 | pwsplusgval.p | . . . 4 ⊢ ✚ = (+g‘𝑌) | |
30 | 13 | fveq2d 6667 | . . . 4 ⊢ (𝜑 → (+g‘𝑌) = (+g‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
31 | 29, 30 | syl5eq 2865 | . . 3 ⊢ (𝜑 → ✚ = (+g‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
32 | 31 | oveqd 7162 | . 2 ⊢ (𝜑 → (𝐹 ✚ 𝐺) = (𝐹(+g‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))𝐺)) |
33 | fvexd 6678 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ V) | |
34 | fvexd 6678 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) ∈ V) | |
35 | eqid 2818 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
36 | 10, 35, 9, 5, 4, 8 | pwselbas 16750 | . . . 4 ⊢ (𝜑 → 𝐹:𝐼⟶(Base‘𝑅)) |
37 | 36 | feqmptd 6726 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥))) |
38 | 10, 35, 9, 5, 4, 17 | pwselbas 16750 | . . . 4 ⊢ (𝜑 → 𝐺:𝐼⟶(Base‘𝑅)) |
39 | 38 | feqmptd 6726 | . . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐼 ↦ (𝐺‘𝑥))) |
40 | 4, 33, 34, 37, 39 | offval2 7415 | . 2 ⊢ (𝜑 → (𝐹 ∘f + 𝐺) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥) + (𝐺‘𝑥)))) |
41 | 28, 32, 40 | 3eqtr4d 2863 | 1 ⊢ (𝜑 → (𝐹 ✚ 𝐺) = (𝐹 ∘f + 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 {csn 4557 ↦ cmpt 5137 × cxp 5546 Fn wfn 6343 ‘cfv 6348 (class class class)co 7145 ∘f cof 7396 Basecbs 16471 +gcplusg 16553 Scalarcsca 16556 Xscprds 16707 ↑s cpws 16708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-hom 16577 df-cco 16578 df-prds 16709 df-pws 16711 |
This theorem is referenced by: pwsdiagmhm 17983 pwsco1mhm 17984 pwsco2mhm 17985 pwssub 18151 pwssplit2 19761 mpfaddcl 20246 mpfind 20248 evl1addd 20432 pf1addcl 20444 frlmplusgval 20836 ply1rem 24684 |
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