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Mirrors > Home > MPE Home > Th. List > Mathboxes > lfladd0l | Structured version Visualization version GIF version |
Description: Functional addition with the zero functional. (Contributed by NM, 21-Oct-2014.) |
Ref | Expression |
---|---|
lfladd0l.v | ⊢ 𝑉 = (Base‘𝑊) |
lfladd0l.r | ⊢ 𝑅 = (Scalar‘𝑊) |
lfladd0l.p | ⊢ + = (+g‘𝑅) |
lfladd0l.o | ⊢ 0 = (0g‘𝑅) |
lfladd0l.f | ⊢ 𝐹 = (LFnl‘𝑊) |
lfladd0l.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lfladd0l.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
Ref | Expression |
---|---|
lfladd0l | ⊢ (𝜑 → ((𝑉 × { 0 }) ∘f + 𝐺) = 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lfladd0l.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
2 | 1 | fvexi 6936 | . . 3 ⊢ 𝑉 ∈ V |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → 𝑉 ∈ V) |
4 | lfladd0l.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
5 | lfladd0l.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
6 | lfladd0l.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑊) | |
7 | eqid 2740 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
8 | lfladd0l.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
9 | 6, 7, 1, 8 | lflf 39021 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶(Base‘𝑅)) |
10 | 4, 5, 9 | syl2anc 583 | . 2 ⊢ (𝜑 → 𝐺:𝑉⟶(Base‘𝑅)) |
11 | lfladd0l.o | . . . 4 ⊢ 0 = (0g‘𝑅) | |
12 | 11 | fvexi 6936 | . . 3 ⊢ 0 ∈ V |
13 | 12 | a1i 11 | . 2 ⊢ (𝜑 → 0 ∈ V) |
14 | 6 | lmodring 20890 | . . . 4 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring) |
15 | ringgrp 20267 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
16 | 4, 14, 15 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Grp) |
17 | lfladd0l.p | . . . 4 ⊢ + = (+g‘𝑅) | |
18 | 7, 17, 11 | grplid 19009 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ 𝑘 ∈ (Base‘𝑅)) → ( 0 + 𝑘) = 𝑘) |
19 | 16, 18 | sylan 579 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑅)) → ( 0 + 𝑘) = 𝑘) |
20 | 3, 10, 13, 19 | caofid0l 7748 | 1 ⊢ (𝜑 → ((𝑉 × { 0 }) ∘f + 𝐺) = 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2108 Vcvv 3488 {csn 4648 × cxp 5698 ⟶wf 6571 ‘cfv 6575 (class class class)co 7450 ∘f cof 7714 Basecbs 17260 +gcplusg 17313 Scalarcsca 17316 0gc0g 17501 Grpcgrp 18975 Ringcrg 20262 LModclmod 20882 LFnlclfn 39015 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-of 7716 df-map 8888 df-0g 17503 df-mgm 18680 df-sgrp 18759 df-mnd 18775 df-grp 18978 df-ring 20264 df-lmod 20884 df-lfl 39016 |
This theorem is referenced by: ldualgrplem 39103 ldual0v 39108 |
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