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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 4077 | . . . . . 6 ⊢ {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin} ⊆ (Base‘(𝑆Xs𝑅)) | |
2 | eqid 2733 | . . . . . . 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 2733 | . . . . . . 7 ⊢ (Base‘(𝑆Xs𝑅)) = (Base‘(𝑆Xs𝑅)) | |
4 | 2, 3 | ressbas2 17179 | . . . . . 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 7417 | . . . 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 2733 | . . . . 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 21281 | . . . 4 ⊢ (𝑅 ∈ V → (𝑆 ⊕m 𝑅) = ((𝑆Xs𝑅) ↾s {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin})) |
9 | 8 | fveq2d 6893 | . . . . 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 7422 | . . . 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 2799 | . . 3 ⊢ (𝑅 ∈ V → (𝑆 ⊕m 𝑅) = ((𝑆Xs𝑅) ↾s (Base‘(𝑆 ⊕m 𝑅)))) |
12 | ress0 17185 | . . . . 5 ⊢ (∅ ↾s (Base‘(𝑆 ⊕m 𝑅))) = ∅ | |
13 | 12 | eqcomi 2742 | . . . 4 ⊢ ∅ = (∅ ↾s (Base‘(𝑆 ⊕m 𝑅))) |
14 | reldmdsmm 21280 | . . . . 5 ⊢ Rel dom ⊕m | |
15 | 14 | ovprc2 7446 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑆 ⊕m 𝑅) = ∅) |
16 | reldmprds 17391 | . . . . . 6 ⊢ Rel dom Xs | |
17 | 16 | ovprc2 7446 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (𝑆Xs𝑅) = ∅) |
18 | 17 | oveq1d 7421 | . . . 4 ⊢ (¬ 𝑅 ∈ V → ((𝑆Xs𝑅) ↾s (Base‘(𝑆 ⊕m 𝑅))) = (∅ ↾s (Base‘(𝑆 ⊕m 𝑅)))) |
19 | 13, 15, 18 | 3eqtr4a 2799 | . . 3 ⊢ (¬ 𝑅 ∈ V → (𝑆 ⊕m 𝑅) = ((𝑆Xs𝑅) ↾s (Base‘(𝑆 ⊕m 𝑅)))) |
20 | 11, 19 | pm2.61i 182 | . 2 ⊢ (𝑆 ⊕m 𝑅) = ((𝑆Xs𝑅) ↾s (Base‘(𝑆 ⊕m 𝑅))) |
21 | dsmmval2.b | . . 3 ⊢ 𝐵 = (Base‘(𝑆 ⊕m 𝑅)) | |
22 | 21 | oveq2i 7417 | . 2 ⊢ ((𝑆Xs𝑅) ↾s 𝐵) = ((𝑆Xs𝑅) ↾s (Base‘(𝑆 ⊕m 𝑅))) |
23 | 20, 22 | eqtr4i 2764 | 1 ⊢ (𝑆 ⊕m 𝑅) = ((𝑆Xs𝑅) ↾s 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 {crab 3433 Vcvv 3475 ⊆ wss 3948 ∅c0 4322 dom cdm 5676 ‘cfv 6541 (class class class)co 7406 Fincfn 8936 Basecbs 17141 ↾s cress 17170 0gc0g 17382 Xscprds 17388 ⊕m cdsmm 21278 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-map 8819 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-mulr 17208 df-sca 17210 df-vsca 17211 df-ip 17212 df-tset 17213 df-ple 17214 df-ds 17216 df-hom 17218 df-cco 17219 df-prds 17390 df-dsmm 21279 |
This theorem is referenced by: dsmmfi 21285 dsmmlmod 21292 frlmpws 21297 |
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