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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 3185 | . 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
2 | oveq12 6558 | . . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (𝑠Xs𝑟) = (𝑆Xs𝑅)) | |
3 | eqid 2610 | . . . . . . . . 9 ⊢ (𝑠Xs𝑟) = (𝑠Xs𝑟) | |
4 | vex 3176 | . . . . . . . . . 10 ⊢ 𝑠 ∈ V | |
5 | 4 | a1i 11 | . . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → 𝑠 ∈ V) |
6 | vex 3176 | . . . . . . . . . 10 ⊢ 𝑟 ∈ V | |
7 | 6 | a1i 11 | . . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → 𝑟 ∈ V) |
8 | eqid 2610 | . . . . . . . . 9 ⊢ (Base‘(𝑠Xs𝑟)) = (Base‘(𝑠Xs𝑟)) | |
9 | eqidd 2611 | . . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → dom 𝑟 = dom 𝑟) | |
10 | 3, 5, 7, 8, 9 | prdsbas 15940 | . . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (Base‘(𝑠Xs𝑟)) = X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥))) |
11 | 2 | fveq2d 6107 | . . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (Base‘(𝑠Xs𝑟)) = (Base‘(𝑆Xs𝑅))) |
12 | 10, 11 | eqtr3d 2646 | . . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) = (Base‘(𝑆Xs𝑅))) |
13 | simpr 476 | . . . . . . . . . 10 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅) | |
14 | 13 | dmeqd 5248 | . . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → dom 𝑟 = dom 𝑅) |
15 | 13 | fveq1d 6105 | . . . . . . . . . . 11 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (𝑟‘𝑥) = (𝑅‘𝑥)) |
16 | 15 | fveq2d 6107 | . . . . . . . . . 10 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (0g‘(𝑟‘𝑥)) = (0g‘(𝑅‘𝑥))) |
17 | 16 | neeq2d 2842 | . . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → ((𝑓‘𝑥) ≠ (0g‘(𝑟‘𝑥)) ↔ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥)))) |
18 | 14, 17 | rabeqbidv 3168 | . . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → {𝑥 ∈ dom 𝑟 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑟‘𝑥))} = {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))}) |
19 | 18 | eleq1d 2672 | . . . . . . 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 3168 | . . . . . 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 | syl6eqr 2662 | . . . . 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 6567 | . . . 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 19895 | . . . 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 6577 | . . . 4 ⊢ ((𝑆Xs𝑅) ↾s 𝐵) ∈ V | |
26 | 23, 24, 25 | ovmpt2a 6689 | . . 3 ⊢ ((𝑆 ∈ V ∧ 𝑅 ∈ V) → (𝑆 ⊕m 𝑅) = ((𝑆Xs𝑅) ↾s 𝐵)) |
27 | reldmdsmm 19896 | . . . . . . 7 ⊢ Rel dom ⊕m | |
28 | 27 | ovprc1 6582 | . . . . . 6 ⊢ (¬ 𝑆 ∈ V → (𝑆 ⊕m 𝑅) = ∅) |
29 | ress0 15761 | . . . . . 6 ⊢ (∅ ↾s 𝐵) = ∅ | |
30 | 28, 29 | syl6eqr 2662 | . . . . 5 ⊢ (¬ 𝑆 ∈ V → (𝑆 ⊕m 𝑅) = (∅ ↾s 𝐵)) |
31 | reldmprds 15932 | . . . . . . 7 ⊢ Rel dom Xs | |
32 | 31 | ovprc1 6582 | . . . . . 6 ⊢ (¬ 𝑆 ∈ V → (𝑆Xs𝑅) = ∅) |
33 | 32 | oveq1d 6564 | . . . . 5 ⊢ (¬ 𝑆 ∈ V → ((𝑆Xs𝑅) ↾s 𝐵) = (∅ ↾s 𝐵)) |
34 | 30, 33 | eqtr4d 2647 | . . . 4 ⊢ (¬ 𝑆 ∈ V → (𝑆 ⊕m 𝑅) = ((𝑆Xs𝑅) ↾s 𝐵)) |
35 | 34 | adantr 480 | . . 3 ⊢ ((¬ 𝑆 ∈ V ∧ 𝑅 ∈ V) → (𝑆 ⊕m 𝑅) = ((𝑆Xs𝑅) ↾s 𝐵)) |
36 | 26, 35 | pm2.61ian 827 | . 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 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 {crab 2900 Vcvv 3173 ∅c0 3874 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 Xcixp 7794 Fincfn 7841 Basecbs 15695 ↾s cress 15696 0gc0g 15923 Xscprds 15929 ⊕m cdsmm 19894 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-hom 15793 df-cco 15794 df-prds 15931 df-dsmm 19895 |
This theorem is referenced by: dsmmbase 19898 dsmmval2 19899 |
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