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Mirrors > Home > MPE Home > Th. List > uvcff | Structured version Visualization version GIF version |
Description: Domain and range of the unit vector generator; ring condition required to be sure 1 and 0 are actually in the ring. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.) |
Ref | Expression |
---|---|
uvcff.u | ⊢ 𝑈 = (𝑅 unitVec 𝐼) |
uvcff.y | ⊢ 𝑌 = (𝑅 freeLMod 𝐼) |
uvcff.b | ⊢ 𝐵 = (Base‘𝑌) |
Ref | Expression |
---|---|
uvcff | ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝑈:𝐼⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uvcff.u | . . 3 ⊢ 𝑈 = (𝑅 unitVec 𝐼) | |
2 | eqid 2821 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
3 | eqid 2821 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
4 | 1, 2, 3 | uvcfval 20928 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝑈 = (𝑖 ∈ 𝐼 ↦ (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))))) |
5 | eqid 2821 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | 5, 2 | ringidcl 19318 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅)) |
7 | 5, 3 | ring0cl 19319 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅)) |
8 | 6, 7 | ifcld 4512 | . . . . . 6 ⊢ (𝑅 ∈ Ring → if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅)) ∈ (Base‘𝑅)) |
9 | 8 | ad3antrrr 728 | . . . . 5 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅)) ∈ (Base‘𝑅)) |
10 | 9 | fmpttd 6879 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ 𝑖 ∈ 𝐼) → (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))):𝐼⟶(Base‘𝑅)) |
11 | fvex 6683 | . . . . . 6 ⊢ (Base‘𝑅) ∈ V | |
12 | elmapg 8419 | . . . . . 6 ⊢ (((Base‘𝑅) ∈ V ∧ 𝐼 ∈ 𝑊) → ((𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))) ∈ ((Base‘𝑅) ↑m 𝐼) ↔ (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))):𝐼⟶(Base‘𝑅))) | |
13 | 11, 12 | mpan 688 | . . . . 5 ⊢ (𝐼 ∈ 𝑊 → ((𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))) ∈ ((Base‘𝑅) ↑m 𝐼) ↔ (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))):𝐼⟶(Base‘𝑅))) |
14 | 13 | ad2antlr 725 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ 𝑖 ∈ 𝐼) → ((𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))) ∈ ((Base‘𝑅) ↑m 𝐼) ↔ (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))):𝐼⟶(Base‘𝑅))) |
15 | 10, 14 | mpbird 259 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ 𝑖 ∈ 𝐼) → (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))) ∈ ((Base‘𝑅) ↑m 𝐼)) |
16 | mptexg 6984 | . . . . 5 ⊢ (𝐼 ∈ 𝑊 → (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))) ∈ V) | |
17 | 16 | ad2antlr 725 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ 𝑖 ∈ 𝐼) → (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))) ∈ V) |
18 | funmpt 6393 | . . . . 5 ⊢ Fun (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))) | |
19 | 18 | a1i 11 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ 𝑖 ∈ 𝐼) → Fun (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅)))) |
20 | fvex 6683 | . . . . 5 ⊢ (0g‘𝑅) ∈ V | |
21 | 20 | a1i 11 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ 𝑖 ∈ 𝐼) → (0g‘𝑅) ∈ V) |
22 | snfi 8594 | . . . . 5 ⊢ {𝑖} ∈ Fin | |
23 | 22 | a1i 11 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ 𝑖 ∈ 𝐼) → {𝑖} ∈ Fin) |
24 | eldifsni 4722 | . . . . . . . 8 ⊢ (𝑗 ∈ (𝐼 ∖ {𝑖}) → 𝑗 ≠ 𝑖) | |
25 | 24 | adantl 484 | . . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ (𝐼 ∖ {𝑖})) → 𝑗 ≠ 𝑖) |
26 | 25 | neneqd 3021 | . . . . . 6 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ (𝐼 ∖ {𝑖})) → ¬ 𝑗 = 𝑖) |
27 | 26 | iffalsed 4478 | . . . . 5 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ (𝐼 ∖ {𝑖})) → if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅)) = (0g‘𝑅)) |
28 | simplr 767 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ 𝑖 ∈ 𝐼) → 𝐼 ∈ 𝑊) | |
29 | 27, 28 | suppss2 7864 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ 𝑖 ∈ 𝐼) → ((𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))) supp (0g‘𝑅)) ⊆ {𝑖}) |
30 | suppssfifsupp 8848 | . . . 4 ⊢ ((((𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))) ∈ V ∧ Fun (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))) ∧ (0g‘𝑅) ∈ V) ∧ ({𝑖} ∈ Fin ∧ ((𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))) supp (0g‘𝑅)) ⊆ {𝑖})) → (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))) finSupp (0g‘𝑅)) | |
31 | 17, 19, 21, 23, 29, 30 | syl32anc 1374 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ 𝑖 ∈ 𝐼) → (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))) finSupp (0g‘𝑅)) |
32 | uvcff.y | . . . . 5 ⊢ 𝑌 = (𝑅 freeLMod 𝐼) | |
33 | uvcff.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑌) | |
34 | 32, 5, 3, 33 | frlmelbas 20900 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → ((𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))) ∈ 𝐵 ↔ ((𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))) ∈ ((Base‘𝑅) ↑m 𝐼) ∧ (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))) finSupp (0g‘𝑅)))) |
35 | 34 | adantr 483 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ 𝑖 ∈ 𝐼) → ((𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))) ∈ 𝐵 ↔ ((𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))) ∈ ((Base‘𝑅) ↑m 𝐼) ∧ (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))) finSupp (0g‘𝑅)))) |
36 | 15, 31, 35 | mpbir2and 711 | . 2 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ 𝑖 ∈ 𝐼) → (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))) ∈ 𝐵) |
37 | 4, 36 | fmpt3d 6880 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝑈:𝐼⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 Vcvv 3494 ∖ cdif 3933 ⊆ wss 3936 ifcif 4467 {csn 4567 class class class wbr 5066 ↦ cmpt 5146 Fun wfun 6349 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 supp csupp 7830 ↑m cmap 8406 Fincfn 8509 finSupp cfsupp 8833 Basecbs 16483 0gc0g 16713 1rcur 19251 Ringcrg 19297 freeLMod cfrlm 20890 unitVec cuvc 20926 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-hom 16589 df-cco 16590 df-0g 16715 df-prds 16721 df-pws 16723 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-mgp 19240 df-ur 19252 df-ring 19299 df-sra 19944 df-rgmod 19945 df-dsmm 20876 df-frlm 20891 df-uvc 20927 |
This theorem is referenced by: uvcf1 20936 uvcresum 20937 frlmssuvc1 20938 frlmssuvc2 20939 frlmsslsp 20940 frlmlbs 20941 frlmup2 20943 frlmup3 20944 frlmup4 20945 lindsdom 34901 matunitlindflem2 34904 uvccl 39170 aacllem 44922 |
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