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Mirrors > Home > MPE Home > Th. List > mpofrlmd | Structured version Visualization version GIF version |
Description: Elements of the free module are mappings with two arguments defined by their operation values. (Contributed by AV, 20-Feb-2019.) (Proof shortened by AV, 3-Jul-2022.) |
Ref | Expression |
---|---|
mpofrlmd.f | ⊢ 𝐹 = (𝑅 freeLMod (𝑁 × 𝑀)) |
mpofrlmd.v | ⊢ 𝑉 = (Base‘𝐹) |
mpofrlmd.s | ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → 𝐴 = 𝐵) |
mpofrlmd.a | ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑀) → 𝐴 ∈ 𝑋) |
mpofrlmd.b | ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑀) → 𝐵 ∈ 𝑌) |
mpofrlmd.e | ⊢ (𝜑 → (𝑁 ∈ 𝑈 ∧ 𝑀 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉)) |
Ref | Expression |
---|---|
mpofrlmd | ⊢ (𝜑 → (𝑍 = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑀 ↦ 𝐵) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑀 (𝑖𝑍𝑗) = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpofrlmd.e | . . . 4 ⊢ (𝜑 → (𝑁 ∈ 𝑈 ∧ 𝑀 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉)) | |
2 | xpexg 7466 | . . . . . 6 ⊢ ((𝑁 ∈ 𝑈 ∧ 𝑀 ∈ 𝑊) → (𝑁 × 𝑀) ∈ V) | |
3 | 2 | anim1i 616 | . . . . 5 ⊢ (((𝑁 ∈ 𝑈 ∧ 𝑀 ∈ 𝑊) ∧ 𝑍 ∈ 𝑉) → ((𝑁 × 𝑀) ∈ V ∧ 𝑍 ∈ 𝑉)) |
4 | 3 | 3impa 1105 | . . . 4 ⊢ ((𝑁 ∈ 𝑈 ∧ 𝑀 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → ((𝑁 × 𝑀) ∈ V ∧ 𝑍 ∈ 𝑉)) |
5 | 1, 4 | syl 17 | . . 3 ⊢ (𝜑 → ((𝑁 × 𝑀) ∈ V ∧ 𝑍 ∈ 𝑉)) |
6 | mpofrlmd.f | . . . 4 ⊢ 𝐹 = (𝑅 freeLMod (𝑁 × 𝑀)) | |
7 | eqid 2820 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
8 | mpofrlmd.v | . . . 4 ⊢ 𝑉 = (Base‘𝐹) | |
9 | 6, 7, 8 | frlmbasf 20899 | . . 3 ⊢ (((𝑁 × 𝑀) ∈ V ∧ 𝑍 ∈ 𝑉) → 𝑍:(𝑁 × 𝑀)⟶(Base‘𝑅)) |
10 | ffn 6507 | . . 3 ⊢ (𝑍:(𝑁 × 𝑀)⟶(Base‘𝑅) → 𝑍 Fn (𝑁 × 𝑀)) | |
11 | 5, 9, 10 | 3syl 18 | . 2 ⊢ (𝜑 → 𝑍 Fn (𝑁 × 𝑀)) |
12 | mpofrlmd.s | . 2 ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → 𝐴 = 𝐵) | |
13 | mpofrlmd.a | . 2 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑀) → 𝐴 ∈ 𝑋) | |
14 | mpofrlmd.b | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑀) → 𝐵 ∈ 𝑌) | |
15 | 11, 12, 13, 14 | fnmpoovd 7775 | 1 ⊢ (𝜑 → (𝑍 = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑀 ↦ 𝐵) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑀 (𝑖𝑍𝑗) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 ∀wral 3137 Vcvv 3491 × cxp 5546 Fn wfn 6343 ⟶wf 6344 ‘cfv 6348 (class class class)co 7149 ∈ cmpo 7151 Basecbs 16478 freeLMod cfrlm 20885 |
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 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-supp 7824 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-ixp 8455 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-fsupp 8827 df-sup 8899 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12890 df-struct 16480 df-ndx 16481 df-slot 16482 df-base 16484 df-sets 16485 df-ress 16486 df-plusg 16573 df-mulr 16574 df-sca 16576 df-vsca 16577 df-ip 16578 df-tset 16579 df-ple 16580 df-ds 16582 df-hom 16584 df-cco 16585 df-0g 16710 df-prds 16716 df-pws 16718 df-sra 19939 df-rgmod 19940 df-dsmm 20871 df-frlm 20886 |
This theorem is referenced by: (None) |
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