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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 3515 | . 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
2 | oveq12 7168 | . . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (𝑠Xs𝑟) = (𝑆Xs𝑅)) | |
3 | eqid 2824 | . . . . . . . . 9 ⊢ (𝑠Xs𝑟) = (𝑠Xs𝑟) | |
4 | vex 3500 | . . . . . . . . . 10 ⊢ 𝑠 ∈ V | |
5 | 4 | a1i 11 | . . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → 𝑠 ∈ V) |
6 | vex 3500 | . . . . . . . . . 10 ⊢ 𝑟 ∈ V | |
7 | 6 | a1i 11 | . . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → 𝑟 ∈ V) |
8 | eqid 2824 | . . . . . . . . 9 ⊢ (Base‘(𝑠Xs𝑟)) = (Base‘(𝑠Xs𝑟)) | |
9 | eqidd 2825 | . . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → dom 𝑟 = dom 𝑟) | |
10 | 3, 5, 7, 8, 9 | prdsbas 16733 | . . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (Base‘(𝑠Xs𝑟)) = X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥))) |
11 | 2 | fveq2d 6677 | . . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (Base‘(𝑠Xs𝑟)) = (Base‘(𝑆Xs𝑅))) |
12 | 10, 11 | eqtr3d 2861 | . . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) = (Base‘(𝑆Xs𝑅))) |
13 | simpr 487 | . . . . . . . . . 10 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅) | |
14 | 13 | dmeqd 5777 | . . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → dom 𝑟 = dom 𝑅) |
15 | 13 | fveq1d 6675 | . . . . . . . . . . 11 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (𝑟‘𝑥) = (𝑅‘𝑥)) |
16 | 15 | fveq2d 6677 | . . . . . . . . . 10 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (0g‘(𝑟‘𝑥)) = (0g‘(𝑅‘𝑥))) |
17 | 16 | neeq2d 3079 | . . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → ((𝑓‘𝑥) ≠ (0g‘(𝑟‘𝑥)) ↔ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥)))) |
18 | 14, 17 | rabeqbidv 3488 | . . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → {𝑥 ∈ dom 𝑟 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑟‘𝑥))} = {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))}) |
19 | 18 | eleq1d 2900 | . . . . . . 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 3488 | . . . . . 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 2877 | . . . . 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 7177 | . . . 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 20879 | . . . 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 7192 | . . . 4 ⊢ ((𝑆Xs𝑅) ↾s 𝐵) ∈ V | |
26 | 23, 24, 25 | ovmpoa 7308 | . . 3 ⊢ ((𝑆 ∈ V ∧ 𝑅 ∈ V) → (𝑆 ⊕m 𝑅) = ((𝑆Xs𝑅) ↾s 𝐵)) |
27 | reldmdsmm 20880 | . . . . . . 7 ⊢ Rel dom ⊕m | |
28 | 27 | ovprc1 7198 | . . . . . 6 ⊢ (¬ 𝑆 ∈ V → (𝑆 ⊕m 𝑅) = ∅) |
29 | ress0 16561 | . . . . . 6 ⊢ (∅ ↾s 𝐵) = ∅ | |
30 | 28, 29 | syl6eqr 2877 | . . . . 5 ⊢ (¬ 𝑆 ∈ V → (𝑆 ⊕m 𝑅) = (∅ ↾s 𝐵)) |
31 | reldmprds 16725 | . . . . . . 7 ⊢ Rel dom Xs | |
32 | 31 | ovprc1 7198 | . . . . . 6 ⊢ (¬ 𝑆 ∈ V → (𝑆Xs𝑅) = ∅) |
33 | 32 | oveq1d 7174 | . . . . 5 ⊢ (¬ 𝑆 ∈ V → ((𝑆Xs𝑅) ↾s 𝐵) = (∅ ↾s 𝐵)) |
34 | 30, 33 | eqtr4d 2862 | . . . 4 ⊢ (¬ 𝑆 ∈ V → (𝑆 ⊕m 𝑅) = ((𝑆Xs𝑅) ↾s 𝐵)) |
35 | 34 | adantr 483 | . . 3 ⊢ ((¬ 𝑆 ∈ V ∧ 𝑅 ∈ V) → (𝑆 ⊕m 𝑅) = ((𝑆Xs𝑅) ↾s 𝐵)) |
36 | 26, 35 | pm2.61ian 810 | . 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 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 {crab 3145 Vcvv 3497 ∅c0 4294 dom cdm 5558 ‘cfv 6358 (class class class)co 7159 Xcixp 8464 Fincfn 8512 Basecbs 16486 ↾s cress 16487 0gc0g 16716 Xscprds 16722 ⊕m cdsmm 20878 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-sup 8909 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-ress 16494 df-plusg 16581 df-mulr 16582 df-sca 16584 df-vsca 16585 df-ip 16586 df-tset 16587 df-ple 16588 df-ds 16590 df-hom 16592 df-cco 16593 df-prds 16724 df-dsmm 20879 |
This theorem is referenced by: dsmmbase 20882 dsmmval2 20883 |
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