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Mirrors > Home > MPE Home > Th. List > dsmmelbas | Structured version Visualization version GIF version |
Description: Membership in the finitely supported hull of a structure product in terms of the index set. (Contributed by Stefan O'Rear, 11-Jan-2015.) |
Ref | Expression |
---|---|
dsmmelbas.p | ⊢ 𝑃 = (𝑆Xs𝑅) |
dsmmelbas.c | ⊢ 𝐶 = (𝑆 ⊕m 𝑅) |
dsmmelbas.b | ⊢ 𝐵 = (Base‘𝑃) |
dsmmelbas.h | ⊢ 𝐻 = (Base‘𝐶) |
dsmmelbas.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
dsmmelbas.r | ⊢ (𝜑 → 𝑅 Fn 𝐼) |
Ref | Expression |
---|---|
dsmmelbas | ⊢ (𝜑 → (𝑋 ∈ 𝐻 ↔ (𝑋 ∈ 𝐵 ∧ {𝑎 ∈ 𝐼 ∣ (𝑋‘𝑎) ≠ (0g‘(𝑅‘𝑎))} ∈ Fin))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dsmmelbas.r | . . . . . 6 ⊢ (𝜑 → 𝑅 Fn 𝐼) | |
2 | dsmmelbas.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
3 | fnex 6625 | . . . . . 6 ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) → 𝑅 ∈ V) | |
4 | 1, 2, 3 | syl2anc 573 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ V) |
5 | eqid 2771 | . . . . . 6 ⊢ {𝑏 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑎 ∈ dom 𝑅 ∣ (𝑏‘𝑎) ≠ (0g‘(𝑅‘𝑎))} ∈ Fin} = {𝑏 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑎 ∈ dom 𝑅 ∣ (𝑏‘𝑎) ≠ (0g‘(𝑅‘𝑎))} ∈ Fin} | |
6 | 5 | dsmmbase 20296 | . . . . 5 ⊢ (𝑅 ∈ V → {𝑏 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑎 ∈ dom 𝑅 ∣ (𝑏‘𝑎) ≠ (0g‘(𝑅‘𝑎))} ∈ Fin} = (Base‘(𝑆 ⊕m 𝑅))) |
7 | 4, 6 | syl 17 | . . . 4 ⊢ (𝜑 → {𝑏 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑎 ∈ dom 𝑅 ∣ (𝑏‘𝑎) ≠ (0g‘(𝑅‘𝑎))} ∈ Fin} = (Base‘(𝑆 ⊕m 𝑅))) |
8 | dsmmelbas.h | . . . . 5 ⊢ 𝐻 = (Base‘𝐶) | |
9 | dsmmelbas.c | . . . . . 6 ⊢ 𝐶 = (𝑆 ⊕m 𝑅) | |
10 | 9 | fveq2i 6335 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘(𝑆 ⊕m 𝑅)) |
11 | 8, 10 | eqtri 2793 | . . . 4 ⊢ 𝐻 = (Base‘(𝑆 ⊕m 𝑅)) |
12 | 7, 11 | syl6reqr 2824 | . . 3 ⊢ (𝜑 → 𝐻 = {𝑏 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑎 ∈ dom 𝑅 ∣ (𝑏‘𝑎) ≠ (0g‘(𝑅‘𝑎))} ∈ Fin}) |
13 | 12 | eleq2d 2836 | . 2 ⊢ (𝜑 → (𝑋 ∈ 𝐻 ↔ 𝑋 ∈ {𝑏 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑎 ∈ dom 𝑅 ∣ (𝑏‘𝑎) ≠ (0g‘(𝑅‘𝑎))} ∈ Fin})) |
14 | fveq1 6331 | . . . . . . 7 ⊢ (𝑏 = 𝑋 → (𝑏‘𝑎) = (𝑋‘𝑎)) | |
15 | 14 | neeq1d 3002 | . . . . . 6 ⊢ (𝑏 = 𝑋 → ((𝑏‘𝑎) ≠ (0g‘(𝑅‘𝑎)) ↔ (𝑋‘𝑎) ≠ (0g‘(𝑅‘𝑎)))) |
16 | 15 | rabbidv 3339 | . . . . 5 ⊢ (𝑏 = 𝑋 → {𝑎 ∈ dom 𝑅 ∣ (𝑏‘𝑎) ≠ (0g‘(𝑅‘𝑎))} = {𝑎 ∈ dom 𝑅 ∣ (𝑋‘𝑎) ≠ (0g‘(𝑅‘𝑎))}) |
17 | 16 | eleq1d 2835 | . . . 4 ⊢ (𝑏 = 𝑋 → ({𝑎 ∈ dom 𝑅 ∣ (𝑏‘𝑎) ≠ (0g‘(𝑅‘𝑎))} ∈ Fin ↔ {𝑎 ∈ dom 𝑅 ∣ (𝑋‘𝑎) ≠ (0g‘(𝑅‘𝑎))} ∈ Fin)) |
18 | 17 | elrab 3515 | . . 3 ⊢ (𝑋 ∈ {𝑏 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑎 ∈ dom 𝑅 ∣ (𝑏‘𝑎) ≠ (0g‘(𝑅‘𝑎))} ∈ Fin} ↔ (𝑋 ∈ (Base‘(𝑆Xs𝑅)) ∧ {𝑎 ∈ dom 𝑅 ∣ (𝑋‘𝑎) ≠ (0g‘(𝑅‘𝑎))} ∈ Fin)) |
19 | dsmmelbas.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑃) | |
20 | dsmmelbas.p | . . . . . . . 8 ⊢ 𝑃 = (𝑆Xs𝑅) | |
21 | 20 | fveq2i 6335 | . . . . . . 7 ⊢ (Base‘𝑃) = (Base‘(𝑆Xs𝑅)) |
22 | 19, 21 | eqtr2i 2794 | . . . . . 6 ⊢ (Base‘(𝑆Xs𝑅)) = 𝐵 |
23 | 22 | eleq2i 2842 | . . . . 5 ⊢ (𝑋 ∈ (Base‘(𝑆Xs𝑅)) ↔ 𝑋 ∈ 𝐵) |
24 | 23 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ (Base‘(𝑆Xs𝑅)) ↔ 𝑋 ∈ 𝐵)) |
25 | fndm 6130 | . . . . . 6 ⊢ (𝑅 Fn 𝐼 → dom 𝑅 = 𝐼) | |
26 | rabeq 3342 | . . . . . 6 ⊢ (dom 𝑅 = 𝐼 → {𝑎 ∈ dom 𝑅 ∣ (𝑋‘𝑎) ≠ (0g‘(𝑅‘𝑎))} = {𝑎 ∈ 𝐼 ∣ (𝑋‘𝑎) ≠ (0g‘(𝑅‘𝑎))}) | |
27 | 1, 25, 26 | 3syl 18 | . . . . 5 ⊢ (𝜑 → {𝑎 ∈ dom 𝑅 ∣ (𝑋‘𝑎) ≠ (0g‘(𝑅‘𝑎))} = {𝑎 ∈ 𝐼 ∣ (𝑋‘𝑎) ≠ (0g‘(𝑅‘𝑎))}) |
28 | 27 | eleq1d 2835 | . . . 4 ⊢ (𝜑 → ({𝑎 ∈ dom 𝑅 ∣ (𝑋‘𝑎) ≠ (0g‘(𝑅‘𝑎))} ∈ Fin ↔ {𝑎 ∈ 𝐼 ∣ (𝑋‘𝑎) ≠ (0g‘(𝑅‘𝑎))} ∈ Fin)) |
29 | 24, 28 | anbi12d 616 | . . 3 ⊢ (𝜑 → ((𝑋 ∈ (Base‘(𝑆Xs𝑅)) ∧ {𝑎 ∈ dom 𝑅 ∣ (𝑋‘𝑎) ≠ (0g‘(𝑅‘𝑎))} ∈ Fin) ↔ (𝑋 ∈ 𝐵 ∧ {𝑎 ∈ 𝐼 ∣ (𝑋‘𝑎) ≠ (0g‘(𝑅‘𝑎))} ∈ Fin))) |
30 | 18, 29 | syl5bb 272 | . 2 ⊢ (𝜑 → (𝑋 ∈ {𝑏 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑎 ∈ dom 𝑅 ∣ (𝑏‘𝑎) ≠ (0g‘(𝑅‘𝑎))} ∈ Fin} ↔ (𝑋 ∈ 𝐵 ∧ {𝑎 ∈ 𝐼 ∣ (𝑋‘𝑎) ≠ (0g‘(𝑅‘𝑎))} ∈ Fin))) |
31 | 13, 30 | bitrd 268 | 1 ⊢ (𝜑 → (𝑋 ∈ 𝐻 ↔ (𝑋 ∈ 𝐵 ∧ {𝑎 ∈ 𝐼 ∣ (𝑋‘𝑎) ≠ (0g‘(𝑅‘𝑎))} ∈ Fin))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 {crab 3065 Vcvv 3351 dom cdm 5249 Fn wfn 6026 ‘cfv 6031 (class class class)co 6793 Fincfn 8109 Basecbs 16064 0gc0g 16308 Xscprds 16314 ⊕m cdsmm 20292 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-oadd 7717 df-er 7896 df-map 8011 df-ixp 8063 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-sup 8504 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-5 11284 df-6 11285 df-7 11286 df-8 11287 df-9 11288 df-n0 11495 df-z 11580 df-dec 11696 df-uz 11889 df-fz 12534 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-mulr 16163 df-sca 16165 df-vsca 16166 df-ip 16167 df-tset 16168 df-ple 16169 df-ds 16172 df-hom 16174 df-cco 16175 df-prds 16316 df-dsmm 20293 |
This theorem is referenced by: dsmm0cl 20301 dsmmacl 20302 dsmmsubg 20304 dsmmlss 20305 |
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