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Mirrors > Home > MPE Home > Th. List > lmodfopnelem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for lmodfopne 20459. (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) |
Ref | Expression |
---|---|
lmodfopnelem1 | ⊢ ((𝑊 ∈ LMod ∧ + = · ) → 𝑉 = 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodfopne.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
2 | lmodfopne.s | . . . . 5 ⊢ 𝑆 = (Scalar‘𝑊) | |
3 | lmodfopne.k | . . . . 5 ⊢ 𝐾 = (Base‘𝑆) | |
4 | lmodfopne.t | . . . . 5 ⊢ · = ( ·sf ‘𝑊) | |
5 | 1, 2, 3, 4 | lmodscaf 20443 | . . . 4 ⊢ (𝑊 ∈ LMod → · :(𝐾 × 𝑉)⟶𝑉) |
6 | 5 | ffnd 6705 | . . 3 ⊢ (𝑊 ∈ LMod → · Fn (𝐾 × 𝑉)) |
7 | lmodfopne.a | . . . . 5 ⊢ + = (+𝑓‘𝑊) | |
8 | 1, 7 | plusffn 18552 | . . . 4 ⊢ + Fn (𝑉 × 𝑉) |
9 | fneq1 6629 | . . . . . . . . . . 11 ⊢ ( + = · → ( + Fn (𝑉 × 𝑉) ↔ · Fn (𝑉 × 𝑉))) | |
10 | fndmu 6645 | . . . . . . . . . . . 12 ⊢ (( · Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → (𝑉 × 𝑉) = (𝐾 × 𝑉)) | |
11 | 10 | ex 413 | . . . . . . . . . . 11 ⊢ ( · Fn (𝑉 × 𝑉) → ( · Fn (𝐾 × 𝑉) → (𝑉 × 𝑉) = (𝐾 × 𝑉))) |
12 | 9, 11 | syl6bi 252 | . . . . . . . . . 10 ⊢ ( + = · → ( + Fn (𝑉 × 𝑉) → ( · Fn (𝐾 × 𝑉) → (𝑉 × 𝑉) = (𝐾 × 𝑉)))) |
13 | 12 | com13 88 | . . . . . . . . 9 ⊢ ( · Fn (𝐾 × 𝑉) → ( + Fn (𝑉 × 𝑉) → ( + = · → (𝑉 × 𝑉) = (𝐾 × 𝑉)))) |
14 | 13 | impcom 408 | . . . . . . . 8 ⊢ (( + Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → ( + = · → (𝑉 × 𝑉) = (𝐾 × 𝑉))) |
15 | 1 | lmodbn0 20431 | . . . . . . . . . . 11 ⊢ (𝑊 ∈ LMod → 𝑉 ≠ ∅) |
16 | xp11 6163 | . . . . . . . . . . 11 ⊢ ((𝑉 ≠ ∅ ∧ 𝑉 ≠ ∅) → ((𝑉 × 𝑉) = (𝐾 × 𝑉) ↔ (𝑉 = 𝐾 ∧ 𝑉 = 𝑉))) | |
17 | 15, 15, 16 | syl2anc 584 | . . . . . . . . . 10 ⊢ (𝑊 ∈ LMod → ((𝑉 × 𝑉) = (𝐾 × 𝑉) ↔ (𝑉 = 𝐾 ∧ 𝑉 = 𝑉))) |
18 | 17 | simprbda 499 | . . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ (𝑉 × 𝑉) = (𝐾 × 𝑉)) → 𝑉 = 𝐾) |
19 | 18 | expcom 414 | . . . . . . . 8 ⊢ ((𝑉 × 𝑉) = (𝐾 × 𝑉) → (𝑊 ∈ LMod → 𝑉 = 𝐾)) |
20 | 14, 19 | syl6 35 | . . . . . . 7 ⊢ (( + Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → ( + = · → (𝑊 ∈ LMod → 𝑉 = 𝐾))) |
21 | 20 | com23 86 | . . . . . 6 ⊢ (( + Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → (𝑊 ∈ LMod → ( + = · → 𝑉 = 𝐾))) |
22 | 21 | ex 413 | . . . . 5 ⊢ ( + Fn (𝑉 × 𝑉) → ( · Fn (𝐾 × 𝑉) → (𝑊 ∈ LMod → ( + = · → 𝑉 = 𝐾)))) |
23 | 22 | com23 86 | . . . 4 ⊢ ( + Fn (𝑉 × 𝑉) → (𝑊 ∈ LMod → ( · Fn (𝐾 × 𝑉) → ( + = · → 𝑉 = 𝐾)))) |
24 | 8, 23 | ax-mp 5 | . . 3 ⊢ (𝑊 ∈ LMod → ( · Fn (𝐾 × 𝑉) → ( + = · → 𝑉 = 𝐾))) |
25 | 6, 24 | mpd 15 | . 2 ⊢ (𝑊 ∈ LMod → ( + = · → 𝑉 = 𝐾)) |
26 | 25 | imp 407 | 1 ⊢ ((𝑊 ∈ LMod ∧ + = · ) → 𝑉 = 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 ∅c0 4318 × cxp 5667 Fn wfn 6527 ‘cfv 6532 Basecbs 17126 Scalarcsca 17182 +𝑓cplusf 18540 LModclmod 20420 ·sf cscaf 20421 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-1st 7957 df-2nd 7958 df-0g 17369 df-plusf 18542 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-grp 18797 df-lmod 20422 df-scaf 20423 |
This theorem is referenced by: lmodfopnelem2 20458 |
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