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Theorem lmodfopnelem1 20457
Description: Lemma 1 for lmodfopne 20459. (Contributed by AV, 2-Oct-2021.)
Hypotheses
Ref Expression
lmodfopne.t · = ( ·sf𝑊)
lmodfopne.a + = (+𝑓𝑊)
lmodfopne.v 𝑉 = (Base‘𝑊)
lmodfopne.s 𝑆 = (Scalar‘𝑊)
lmodfopne.k 𝐾 = (Base‘𝑆)
Assertion
Ref Expression
lmodfopnelem1 ((𝑊 ∈ LMod ∧ + = · ) → 𝑉 = 𝐾)

Proof of Theorem lmodfopnelem1
StepHypRef 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 · = ( ·sf𝑊)
51, 2, 3, 4lmodscaf 20443 . . . 4 (𝑊 ∈ LMod → · :(𝐾 × 𝑉)⟶𝑉)
65ffnd 6705 . . 3 (𝑊 ∈ LMod → · Fn (𝐾 × 𝑉))
7 lmodfopne.a . . . . 5 + = (+𝑓𝑊)
81, 7plusffn 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))
1110ex 413 . . . . . . . . . . 11 ( · Fn (𝑉 × 𝑉) → ( · Fn (𝐾 × 𝑉) → (𝑉 × 𝑉) = (𝐾 × 𝑉)))
129, 11syl6bi 252 . . . . . . . . . 10 ( + = · → ( + Fn (𝑉 × 𝑉) → ( · Fn (𝐾 × 𝑉) → (𝑉 × 𝑉) = (𝐾 × 𝑉))))
1312com13 88 . . . . . . . . 9 ( · Fn (𝐾 × 𝑉) → ( + Fn (𝑉 × 𝑉) → ( + = · → (𝑉 × 𝑉) = (𝐾 × 𝑉))))
1413impcom 408 . . . . . . . 8 (( + Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → ( + = · → (𝑉 × 𝑉) = (𝐾 × 𝑉)))
151lmodbn0 20431 . . . . . . . . . . 11 (𝑊 ∈ LMod → 𝑉 ≠ ∅)
16 xp11 6163 . . . . . . . . . . 11 ((𝑉 ≠ ∅ ∧ 𝑉 ≠ ∅) → ((𝑉 × 𝑉) = (𝐾 × 𝑉) ↔ (𝑉 = 𝐾𝑉 = 𝑉)))
1715, 15, 16syl2anc 584 . . . . . . . . . 10 (𝑊 ∈ LMod → ((𝑉 × 𝑉) = (𝐾 × 𝑉) ↔ (𝑉 = 𝐾𝑉 = 𝑉)))
1817simprbda 499 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ (𝑉 × 𝑉) = (𝐾 × 𝑉)) → 𝑉 = 𝐾)
1918expcom 414 . . . . . . . 8 ((𝑉 × 𝑉) = (𝐾 × 𝑉) → (𝑊 ∈ LMod → 𝑉 = 𝐾))
2014, 19syl6 35 . . . . . . 7 (( + Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → ( + = · → (𝑊 ∈ LMod → 𝑉 = 𝐾)))
2120com23 86 . . . . . 6 (( + Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → (𝑊 ∈ LMod → ( + = ·𝑉 = 𝐾)))
2221ex 413 . . . . 5 ( + Fn (𝑉 × 𝑉) → ( · Fn (𝐾 × 𝑉) → (𝑊 ∈ LMod → ( + = ·𝑉 = 𝐾))))
2322com23 86 . . . 4 ( + Fn (𝑉 × 𝑉) → (𝑊 ∈ LMod → ( · Fn (𝐾 × 𝑉) → ( + = ·𝑉 = 𝐾))))
248, 23ax-mp 5 . . 3 (𝑊 ∈ LMod → ( · Fn (𝐾 × 𝑉) → ( + = ·𝑉 = 𝐾)))
256, 24mpd 15 . 2 (𝑊 ∈ LMod → ( + = ·𝑉 = 𝐾))
2625imp 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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