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Mirrors > Home > MPE Home > Th. List > dsmmval | Structured version Visualization version GIF version |
Description: Value of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.) |
Ref | Expression |
---|---|
dsmmval.b | ⊢ 𝐵 = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin} |
Ref | Expression |
---|---|
dsmmval | ⊢ (𝑅 ∈ 𝑉 → (𝑆 ⊕m 𝑅) = ((𝑆Xs𝑅) ↾s 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3416 | . 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
2 | oveq12 7200 | . . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (𝑠Xs𝑟) = (𝑆Xs𝑅)) | |
3 | eqid 2736 | . . . . . . . . 9 ⊢ (𝑠Xs𝑟) = (𝑠Xs𝑟) | |
4 | vex 3402 | . . . . . . . . . 10 ⊢ 𝑠 ∈ V | |
5 | 4 | a1i 11 | . . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → 𝑠 ∈ V) |
6 | vex 3402 | . . . . . . . . . 10 ⊢ 𝑟 ∈ V | |
7 | 6 | a1i 11 | . . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → 𝑟 ∈ V) |
8 | eqid 2736 | . . . . . . . . 9 ⊢ (Base‘(𝑠Xs𝑟)) = (Base‘(𝑠Xs𝑟)) | |
9 | eqidd 2737 | . . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → dom 𝑟 = dom 𝑟) | |
10 | 3, 5, 7, 8, 9 | prdsbas 16916 | . . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (Base‘(𝑠Xs𝑟)) = X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥))) |
11 | 2 | fveq2d 6699 | . . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (Base‘(𝑠Xs𝑟)) = (Base‘(𝑆Xs𝑅))) |
12 | 10, 11 | eqtr3d 2773 | . . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) = (Base‘(𝑆Xs𝑅))) |
13 | simpr 488 | . . . . . . . . . 10 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅) | |
14 | 13 | dmeqd 5759 | . . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → dom 𝑟 = dom 𝑅) |
15 | 13 | fveq1d 6697 | . . . . . . . . . . 11 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (𝑟‘𝑥) = (𝑅‘𝑥)) |
16 | 15 | fveq2d 6699 | . . . . . . . . . 10 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (0g‘(𝑟‘𝑥)) = (0g‘(𝑅‘𝑥))) |
17 | 16 | neeq2d 2992 | . . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → ((𝑓‘𝑥) ≠ (0g‘(𝑟‘𝑥)) ↔ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥)))) |
18 | 14, 17 | rabeqbidv 3386 | . . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → {𝑥 ∈ dom 𝑟 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑟‘𝑥))} = {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))}) |
19 | 18 | eleq1d 2815 | . . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → ({𝑥 ∈ dom 𝑟 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑟‘𝑥))} ∈ Fin ↔ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin)) |
20 | 12, 19 | rabeqbidv 3386 | . . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → {𝑓 ∈ X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ∣ {𝑥 ∈ dom 𝑟 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑟‘𝑥))} ∈ Fin} = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin}) |
21 | dsmmval.b | . . . . . 6 ⊢ 𝐵 = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin} | |
22 | 20, 21 | eqtr4di 2789 | . . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → {𝑓 ∈ X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ∣ {𝑥 ∈ dom 𝑟 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑟‘𝑥))} ∈ Fin} = 𝐵) |
23 | 2, 22 | oveq12d 7209 | . . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → ((𝑠Xs𝑟) ↾s {𝑓 ∈ X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ∣ {𝑥 ∈ dom 𝑟 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑟‘𝑥))} ∈ Fin}) = ((𝑆Xs𝑅) ↾s 𝐵)) |
24 | df-dsmm 20648 | . . . 4 ⊢ ⊕m = (𝑠 ∈ V, 𝑟 ∈ V ↦ ((𝑠Xs𝑟) ↾s {𝑓 ∈ X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ∣ {𝑥 ∈ dom 𝑟 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑟‘𝑥))} ∈ Fin})) | |
25 | ovex 7224 | . . . 4 ⊢ ((𝑆Xs𝑅) ↾s 𝐵) ∈ V | |
26 | 23, 24, 25 | ovmpoa 7342 | . . 3 ⊢ ((𝑆 ∈ V ∧ 𝑅 ∈ V) → (𝑆 ⊕m 𝑅) = ((𝑆Xs𝑅) ↾s 𝐵)) |
27 | reldmdsmm 20649 | . . . . . . 7 ⊢ Rel dom ⊕m | |
28 | 27 | ovprc1 7230 | . . . . . 6 ⊢ (¬ 𝑆 ∈ V → (𝑆 ⊕m 𝑅) = ∅) |
29 | ress0 16742 | . . . . . 6 ⊢ (∅ ↾s 𝐵) = ∅ | |
30 | 28, 29 | eqtr4di 2789 | . . . . 5 ⊢ (¬ 𝑆 ∈ V → (𝑆 ⊕m 𝑅) = (∅ ↾s 𝐵)) |
31 | reldmprds 16907 | . . . . . . 7 ⊢ Rel dom Xs | |
32 | 31 | ovprc1 7230 | . . . . . 6 ⊢ (¬ 𝑆 ∈ V → (𝑆Xs𝑅) = ∅) |
33 | 32 | oveq1d 7206 | . . . . 5 ⊢ (¬ 𝑆 ∈ V → ((𝑆Xs𝑅) ↾s 𝐵) = (∅ ↾s 𝐵)) |
34 | 30, 33 | eqtr4d 2774 | . . . 4 ⊢ (¬ 𝑆 ∈ V → (𝑆 ⊕m 𝑅) = ((𝑆Xs𝑅) ↾s 𝐵)) |
35 | 34 | adantr 484 | . . 3 ⊢ ((¬ 𝑆 ∈ V ∧ 𝑅 ∈ V) → (𝑆 ⊕m 𝑅) = ((𝑆Xs𝑅) ↾s 𝐵)) |
36 | 26, 35 | pm2.61ian 812 | . 2 ⊢ (𝑅 ∈ V → (𝑆 ⊕m 𝑅) = ((𝑆Xs𝑅) ↾s 𝐵)) |
37 | 1, 36 | syl 17 | 1 ⊢ (𝑅 ∈ 𝑉 → (𝑆 ⊕m 𝑅) = ((𝑆Xs𝑅) ↾s 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 {crab 3055 Vcvv 3398 ∅c0 4223 dom cdm 5536 ‘cfv 6358 (class class class)co 7191 Xcixp 8556 Fincfn 8604 Basecbs 16666 ↾s cress 16667 0gc0g 16898 Xscprds 16904 ⊕m cdsmm 20647 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-sup 9036 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-fz 13061 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-ress 16674 df-plusg 16762 df-mulr 16763 df-sca 16765 df-vsca 16766 df-ip 16767 df-tset 16768 df-ple 16769 df-ds 16771 df-hom 16773 df-cco 16774 df-prds 16906 df-dsmm 20648 |
This theorem is referenced by: dsmmbase 20651 dsmmval2 20652 |
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