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Theorem lmodfopnelem1 19072
 Description: Lemma 1 for lmodfopne 19074. (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 19058 . . . 4 (𝑊 ∈ LMod → · :(𝐾 × 𝑉)⟶𝑉)
65ffnd 6195 . . 3 (𝑊 ∈ LMod → · Fn (𝐾 × 𝑉))
7 lmodfopne.a . . . . 5 + = (+𝑓𝑊)
81, 7plusffn 17422 . . . 4 + Fn (𝑉 × 𝑉)
9 fneq1 6128 . . . . . . . . . . 11 ( + = · → ( + Fn (𝑉 × 𝑉) ↔ · Fn (𝑉 × 𝑉)))
10 fndmu 6141 . . . . . . . . . . . 12 (( · Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → (𝑉 × 𝑉) = (𝐾 × 𝑉))
1110ex 449 . . . . . . . . . . 11 ( · Fn (𝑉 × 𝑉) → ( · Fn (𝐾 × 𝑉) → (𝑉 × 𝑉) = (𝐾 × 𝑉)))
129, 11syl6bi 243 . . . . . . . . . 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 445 . . . . . . . 8 (( + Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → ( + = · → (𝑉 × 𝑉) = (𝐾 × 𝑉)))
151lmodbn0 19046 . . . . . . . . . . 11 (𝑊 ∈ LMod → 𝑉 ≠ ∅)
16 xp11 5715 . . . . . . . . . . 11 ((𝑉 ≠ ∅ ∧ 𝑉 ≠ ∅) → ((𝑉 × 𝑉) = (𝐾 × 𝑉) ↔ (𝑉 = 𝐾𝑉 = 𝑉)))
1715, 15, 16syl2anc 696 . . . . . . . . . 10 (𝑊 ∈ LMod → ((𝑉 × 𝑉) = (𝐾 × 𝑉) ↔ (𝑉 = 𝐾𝑉 = 𝑉)))
1817simprbda 654 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ (𝑉 × 𝑉) = (𝐾 × 𝑉)) → 𝑉 = 𝐾)
1918expcom 450 . . . . . . . 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 449 . . . . 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 444 1 ((𝑊 ∈ LMod ∧ + = · ) → 𝑉 = 𝐾)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1620   ∈ wcel 2127   ≠ wne 2920  ∅c0 4046   × cxp 5252   Fn wfn 6032  ‘cfv 6037  Basecbs 16030  Scalarcsca 16117  +𝑓cplusf 17411  LModclmod 19036   ·sf cscaf 19037 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rmo 3046  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-fv 6045  df-riota 6762  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-1st 7321  df-2nd 7322  df-slot 16034  df-base 16036  df-0g 16275  df-plusf 17413  df-mgm 17414  df-sgrp 17456  df-mnd 17467  df-grp 17597  df-lmod 19038  df-scaf 19039 This theorem is referenced by:  lmodfopnelem2  19073
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