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Mirrors > Home > MPE Home > Th. List > lmodfopnelem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for lmodfopne 20313. (Contributed by AV, 2-Oct-2021.) |
Ref | Expression |
---|---|
lmodfopne.t | ⊢ · = ( ·sf ‘𝑊) |
lmodfopne.a | ⊢ + = (+𝑓‘𝑊) |
lmodfopne.v | ⊢ 𝑉 = (Base‘𝑊) |
lmodfopne.s | ⊢ 𝑆 = (Scalar‘𝑊) |
lmodfopne.k | ⊢ 𝐾 = (Base‘𝑆) |
lmodfopne.0 | ⊢ 0 = (0g‘𝑆) |
lmodfopne.1 | ⊢ 1 = (1r‘𝑆) |
Ref | Expression |
---|---|
lmodfopnelem2 | ⊢ ((𝑊 ∈ LMod ∧ + = · ) → ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodfopne.t | . . . . 5 ⊢ · = ( ·sf ‘𝑊) | |
2 | lmodfopne.a | . . . . 5 ⊢ + = (+𝑓‘𝑊) | |
3 | lmodfopne.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
4 | lmodfopne.s | . . . . 5 ⊢ 𝑆 = (Scalar‘𝑊) | |
5 | lmodfopne.k | . . . . 5 ⊢ 𝐾 = (Base‘𝑆) | |
6 | 1, 2, 3, 4, 5 | lmodfopnelem1 20311 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ + = · ) → 𝑉 = 𝐾) |
7 | 6 | ex 414 | . . 3 ⊢ (𝑊 ∈ LMod → ( + = · → 𝑉 = 𝐾)) |
8 | lmodfopne.0 | . . . . . 6 ⊢ 0 = (0g‘𝑆) | |
9 | 4, 5, 8 | lmod0cl 20301 | . . . . 5 ⊢ (𝑊 ∈ LMod → 0 ∈ 𝐾) |
10 | lmodfopne.1 | . . . . . 6 ⊢ 1 = (1r‘𝑆) | |
11 | 4, 5, 10 | lmod1cl 20302 | . . . . 5 ⊢ (𝑊 ∈ LMod → 1 ∈ 𝐾) |
12 | 9, 11 | jca 513 | . . . 4 ⊢ (𝑊 ∈ LMod → ( 0 ∈ 𝐾 ∧ 1 ∈ 𝐾)) |
13 | eleq2 2827 | . . . . 5 ⊢ (𝑉 = 𝐾 → ( 0 ∈ 𝑉 ↔ 0 ∈ 𝐾)) | |
14 | eleq2 2827 | . . . . 5 ⊢ (𝑉 = 𝐾 → ( 1 ∈ 𝑉 ↔ 1 ∈ 𝐾)) | |
15 | 13, 14 | anbi12d 632 | . . . 4 ⊢ (𝑉 = 𝐾 → (( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉) ↔ ( 0 ∈ 𝐾 ∧ 1 ∈ 𝐾))) |
16 | 12, 15 | syl5ibrcom 247 | . . 3 ⊢ (𝑊 ∈ LMod → (𝑉 = 𝐾 → ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉))) |
17 | 7, 16 | syld 47 | . 2 ⊢ (𝑊 ∈ LMod → ( + = · → ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉))) |
18 | 17 | imp 408 | 1 ⊢ ((𝑊 ∈ LMod ∧ + = · ) → ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ‘cfv 6494 Basecbs 17043 Scalarcsca 17096 0gc0g 17281 +𝑓cplusf 18454 1rcur 19872 LModclmod 20275 ·sf cscaf 20276 |
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 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-cnex 11066 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 ax-pre-mulgt0 11087 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7796 df-1st 7914 df-2nd 7915 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-er 8607 df-en 8843 df-dom 8844 df-sdom 8845 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-sub 11346 df-neg 11347 df-nn 12113 df-2 12175 df-sets 16996 df-slot 17014 df-ndx 17026 df-base 17044 df-plusg 17106 df-0g 17283 df-plusf 18456 df-mgm 18457 df-sgrp 18506 df-mnd 18517 df-grp 18711 df-mgp 19856 df-ur 19873 df-ring 19920 df-lmod 20277 df-scaf 20278 |
This theorem is referenced by: lmodfopne 20313 |
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