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Mirrors > Home > ILE Home > Th. List > lmodfopnelem2 | GIF version |
Description: Lemma 2 for lmodfopne 13825. (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 13823 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ + = · ) → 𝑉 = 𝐾) |
7 | 6 | ex 115 | . . 3 ⊢ (𝑊 ∈ LMod → ( + = · → 𝑉 = 𝐾)) |
8 | lmodfopne.0 | . . . . . 6 ⊢ 0 = (0g‘𝑆) | |
9 | 4, 5, 8 | lmod0cl 13813 | . . . . 5 ⊢ (𝑊 ∈ LMod → 0 ∈ 𝐾) |
10 | lmodfopne.1 | . . . . . 6 ⊢ 1 = (1r‘𝑆) | |
11 | 4, 5, 10 | lmod1cl 13814 | . . . . 5 ⊢ (𝑊 ∈ LMod → 1 ∈ 𝐾) |
12 | 9, 11 | jca 306 | . . . 4 ⊢ (𝑊 ∈ LMod → ( 0 ∈ 𝐾 ∧ 1 ∈ 𝐾)) |
13 | eleq2 2257 | . . . . 5 ⊢ (𝑉 = 𝐾 → ( 0 ∈ 𝑉 ↔ 0 ∈ 𝐾)) | |
14 | eleq2 2257 | . . . . 5 ⊢ (𝑉 = 𝐾 → ( 1 ∈ 𝑉 ↔ 1 ∈ 𝐾)) | |
15 | 13, 14 | anbi12d 473 | . . . 4 ⊢ (𝑉 = 𝐾 → (( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉) ↔ ( 0 ∈ 𝐾 ∧ 1 ∈ 𝐾))) |
16 | 12, 15 | syl5ibrcom 157 | . . 3 ⊢ (𝑊 ∈ LMod → (𝑉 = 𝐾 → ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉))) |
17 | 7, 16 | syld 45 | . 2 ⊢ (𝑊 ∈ LMod → ( + = · → ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉))) |
18 | 17 | imp 124 | 1 ⊢ ((𝑊 ∈ LMod ∧ + = · ) → ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2164 ‘cfv 5255 Basecbs 12621 Scalarcsca 12701 0gc0g 12870 +𝑓cplusf 12939 1rcur 13458 LModclmod 13786 ·sf cscaf 13787 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-addcom 7974 ax-addass 7976 ax-i2m1 7979 ax-0lt1 7980 ax-0id 7982 ax-rnegex 7983 ax-pre-ltirr 7986 ax-pre-ltadd 7990 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-id 4325 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-pnf 8058 df-mnf 8059 df-ltxr 8061 df-inn 8985 df-2 9043 df-3 9044 df-4 9045 df-5 9046 df-6 9047 df-ndx 12624 df-slot 12625 df-base 12627 df-sets 12628 df-plusg 12711 df-mulr 12712 df-sca 12714 df-vsca 12715 df-0g 12872 df-plusf 12941 df-mgm 12942 df-sgrp 12988 df-mnd 13001 df-grp 13078 df-mgp 13420 df-ur 13459 df-ring 13497 df-lmod 13788 df-scaf 13789 |
This theorem is referenced by: lmodfopne 13825 |
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