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Theorem lmodfopnelem1 18947
Description: Lemma 1 for lmodfopne 18949. (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 18933 . . . 4 (𝑊 ∈ LMod → · :(𝐾 × 𝑉)⟶𝑉)
65ffnd 6084 . . 3 (𝑊 ∈ LMod → · Fn (𝐾 × 𝑉))
7 lmodfopne.a . . . . 5 + = (+𝑓𝑊)
81, 7plusffn 17297 . . . 4 + Fn (𝑉 × 𝑉)
9 fneq1 6017 . . . . . . . . . . 11 ( + = · → ( + Fn (𝑉 × 𝑉) ↔ · Fn (𝑉 × 𝑉)))
10 fndmu 6030 . . . . . . . . . . . 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 18921 . . . . . . . . . . 11 (𝑊 ∈ LMod → 𝑉 ≠ ∅)
16 xp11 5604 . . . . . . . . . . 11 ((𝑉 ≠ ∅ ∧ 𝑉 ≠ ∅) → ((𝑉 × 𝑉) = (𝐾 × 𝑉) ↔ (𝑉 = 𝐾𝑉 = 𝑉)))
1715, 15, 16syl2anc 694 . . . . . . . . . 10 (𝑊 ∈ LMod → ((𝑉 × 𝑉) = (𝐾 × 𝑉) ↔ (𝑉 = 𝐾𝑉 = 𝑉)))
1817simprbda 652 . . . . . . . . 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 1523  wcel 2030  wne 2823  c0 3948   × cxp 5141   Fn wfn 5921  cfv 5926  Basecbs 15904  Scalarcsca 15991  +𝑓cplusf 17286  LModclmod 18911   ·sf cscaf 18912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-slot 15908  df-base 15910  df-0g 16149  df-plusf 17288  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-lmod 18913  df-scaf 18914
This theorem is referenced by:  lmodfopnelem2  18948
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