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Mirrors > Home > MPE Home > Th. List > dsmmval2 | Structured version Visualization version GIF version |
Description: Self-referential definition of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.) |
Ref | Expression |
---|---|
dsmmval2.b | ⊢ 𝐵 = (Base‘(𝑆 ⊕m 𝑅)) |
Ref | Expression |
---|---|
dsmmval2 | ⊢ (𝑆 ⊕m 𝑅) = ((𝑆Xs𝑅) ↾s 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 4055 | . . . . . 6 ⊢ {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin} ⊆ (Base‘(𝑆Xs𝑅)) | |
2 | eqid 2821 | . . . . . . 7 ⊢ ((𝑆Xs𝑅) ↾s {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin}) = ((𝑆Xs𝑅) ↾s {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin}) | |
3 | eqid 2821 | . . . . . . 7 ⊢ (Base‘(𝑆Xs𝑅)) = (Base‘(𝑆Xs𝑅)) | |
4 | 2, 3 | ressbas2 16554 | . . . . . 6 ⊢ ({𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin} ⊆ (Base‘(𝑆Xs𝑅)) → {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin} = (Base‘((𝑆Xs𝑅) ↾s {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin}))) |
5 | 1, 4 | ax-mp 5 | . . . . 5 ⊢ {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin} = (Base‘((𝑆Xs𝑅) ↾s {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin})) |
6 | 5 | oveq2i 7166 | . . . 4 ⊢ ((𝑆Xs𝑅) ↾s {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin}) = ((𝑆Xs𝑅) ↾s (Base‘((𝑆Xs𝑅) ↾s {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin}))) |
7 | eqid 2821 | . . . . 5 ⊢ {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin} = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin} | |
8 | 7 | dsmmval 20877 | . . . 4 ⊢ (𝑅 ∈ V → (𝑆 ⊕m 𝑅) = ((𝑆Xs𝑅) ↾s {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin})) |
9 | 8 | fveq2d 6673 | . . . . 5 ⊢ (𝑅 ∈ V → (Base‘(𝑆 ⊕m 𝑅)) = (Base‘((𝑆Xs𝑅) ↾s {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin}))) |
10 | 9 | oveq2d 7171 | . . . 4 ⊢ (𝑅 ∈ V → ((𝑆Xs𝑅) ↾s (Base‘(𝑆 ⊕m 𝑅))) = ((𝑆Xs𝑅) ↾s (Base‘((𝑆Xs𝑅) ↾s {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin})))) |
11 | 6, 8, 10 | 3eqtr4a 2882 | . . 3 ⊢ (𝑅 ∈ V → (𝑆 ⊕m 𝑅) = ((𝑆Xs𝑅) ↾s (Base‘(𝑆 ⊕m 𝑅)))) |
12 | ress0 16557 | . . . . 5 ⊢ (∅ ↾s (Base‘(𝑆 ⊕m 𝑅))) = ∅ | |
13 | 12 | eqcomi 2830 | . . . 4 ⊢ ∅ = (∅ ↾s (Base‘(𝑆 ⊕m 𝑅))) |
14 | reldmdsmm 20876 | . . . . 5 ⊢ Rel dom ⊕m | |
15 | 14 | ovprc2 7195 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑆 ⊕m 𝑅) = ∅) |
16 | reldmprds 16721 | . . . . . 6 ⊢ Rel dom Xs | |
17 | 16 | ovprc2 7195 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (𝑆Xs𝑅) = ∅) |
18 | 17 | oveq1d 7170 | . . . 4 ⊢ (¬ 𝑅 ∈ V → ((𝑆Xs𝑅) ↾s (Base‘(𝑆 ⊕m 𝑅))) = (∅ ↾s (Base‘(𝑆 ⊕m 𝑅)))) |
19 | 13, 15, 18 | 3eqtr4a 2882 | . . 3 ⊢ (¬ 𝑅 ∈ V → (𝑆 ⊕m 𝑅) = ((𝑆Xs𝑅) ↾s (Base‘(𝑆 ⊕m 𝑅)))) |
20 | 11, 19 | pm2.61i 184 | . 2 ⊢ (𝑆 ⊕m 𝑅) = ((𝑆Xs𝑅) ↾s (Base‘(𝑆 ⊕m 𝑅))) |
21 | dsmmval2.b | . . 3 ⊢ 𝐵 = (Base‘(𝑆 ⊕m 𝑅)) | |
22 | 21 | oveq2i 7166 | . 2 ⊢ ((𝑆Xs𝑅) ↾s 𝐵) = ((𝑆Xs𝑅) ↾s (Base‘(𝑆 ⊕m 𝑅))) |
23 | 20, 22 | eqtr4i 2847 | 1 ⊢ (𝑆 ⊕m 𝑅) = ((𝑆Xs𝑅) ↾s 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 {crab 3142 Vcvv 3494 ⊆ wss 3935 ∅c0 4290 dom cdm 5554 ‘cfv 6354 (class class class)co 7155 Fincfn 8508 Basecbs 16482 ↾s cress 16483 0gc0g 16712 Xscprds 16718 ⊕m cdsmm 20874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-sup 8905 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-fz 12892 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-hom 16588 df-cco 16589 df-prds 16720 df-dsmm 20875 |
This theorem is referenced by: dsmmfi 20881 dsmmlmod 20888 frlmpws 20893 |
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