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Mirrors > Home > MPE Home > Th. List > lmodfopnelem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for lmodfopne 19674. (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 19672 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ + = · ) → 𝑉 = 𝐾) |
7 | 6 | ex 415 | . . 3 ⊢ (𝑊 ∈ LMod → ( + = · → 𝑉 = 𝐾)) |
8 | lmodfopne.0 | . . . . . 6 ⊢ 0 = (0g‘𝑆) | |
9 | 4, 5, 8 | lmod0cl 19662 | . . . . 5 ⊢ (𝑊 ∈ LMod → 0 ∈ 𝐾) |
10 | lmodfopne.1 | . . . . . 6 ⊢ 1 = (1r‘𝑆) | |
11 | 4, 5, 10 | lmod1cl 19663 | . . . . 5 ⊢ (𝑊 ∈ LMod → 1 ∈ 𝐾) |
12 | 9, 11 | jca 514 | . . . 4 ⊢ (𝑊 ∈ LMod → ( 0 ∈ 𝐾 ∧ 1 ∈ 𝐾)) |
13 | eleq2 2903 | . . . . 5 ⊢ (𝑉 = 𝐾 → ( 0 ∈ 𝑉 ↔ 0 ∈ 𝐾)) | |
14 | eleq2 2903 | . . . . 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 249 | . . 3 ⊢ (𝑊 ∈ LMod → (𝑉 = 𝐾 → ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉))) |
17 | 7, 16 | syld 47 | . 2 ⊢ (𝑊 ∈ LMod → ( + = · → ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉))) |
18 | 17 | imp 409 | 1 ⊢ ((𝑊 ∈ LMod ∧ + = · ) → ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 Basecbs 16485 Scalarcsca 16570 0gc0g 16715 +𝑓cplusf 17851 1rcur 19253 LModclmod 19636 ·sf cscaf 19637 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-plusg 16580 df-0g 16717 df-plusf 17853 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-mgp 19242 df-ur 19254 df-ring 19301 df-lmod 19638 df-scaf 19639 |
This theorem is referenced by: lmodfopne 19674 |
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