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Theorem lmodfopnelem1 18815
Description: Lemma 1 for lmodfopne 18817. (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 18801 . . . 4 (𝑊 ∈ LMod → · :(𝐾 × 𝑉)⟶𝑉)
65ffnd 6005 . . 3 (𝑊 ∈ LMod → · Fn (𝐾 × 𝑉))
7 lmodfopne.a . . . . 5 + = (+𝑓𝑊)
81, 7plusffn 17166 . . . 4 + Fn (𝑉 × 𝑉)
9 fneq1 5939 . . . . . . . . . . 11 ( + = · → ( + Fn (𝑉 × 𝑉) ↔ · Fn (𝑉 × 𝑉)))
10 fndmu 5952 . . . . . . . . . . . 12 (( · Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → (𝑉 × 𝑉) = (𝐾 × 𝑉))
1110ex 450 . . . . . . . . . . 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 446 . . . . . . . 8 (( + Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → ( + = · → (𝑉 × 𝑉) = (𝐾 × 𝑉)))
151lmodbn0 18789 . . . . . . . . . . 11 (𝑊 ∈ LMod → 𝑉 ≠ ∅)
16 xp11 5532 . . . . . . . . . . 11 ((𝑉 ≠ ∅ ∧ 𝑉 ≠ ∅) → ((𝑉 × 𝑉) = (𝐾 × 𝑉) ↔ (𝑉 = 𝐾𝑉 = 𝑉)))
1715, 15, 16syl2anc 692 . . . . . . . . . 10 (𝑊 ∈ LMod → ((𝑉 × 𝑉) = (𝐾 × 𝑉) ↔ (𝑉 = 𝐾𝑉 = 𝑉)))
1817simprbda 652 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ (𝑉 × 𝑉) = (𝐾 × 𝑉)) → 𝑉 = 𝐾)
1918expcom 451 . . . . . . . 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 450 . . . . 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 445 1 ((𝑊 ∈ LMod ∧ + = · ) → 𝑉 = 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  wne 2796  c0 3896   × cxp 5077   Fn wfn 5845  cfv 5850  Basecbs 15776  Scalarcsca 15860  +𝑓cplusf 17155  LModclmod 18779   ·sf cscaf 18780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117  df-slot 15780  df-base 15781  df-0g 16018  df-plusf 17157  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-grp 17341  df-lmod 18781  df-scaf 18782
This theorem is referenced by:  lmodfopnelem2  18816
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