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Theorem lmodfopnelem1 19666
 Description: Lemma 1 for lmodfopne 19668. (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 19652 . . . 4 (𝑊 ∈ LMod → · :(𝐾 × 𝑉)⟶𝑉)
65ffnd 6488 . . 3 (𝑊 ∈ LMod → · Fn (𝐾 × 𝑉))
7 lmodfopne.a . . . . 5 + = (+𝑓𝑊)
81, 7plusffn 17855 . . . 4 + Fn (𝑉 × 𝑉)
9 fneq1 6414 . . . . . . . . . . 11 ( + = · → ( + Fn (𝑉 × 𝑉) ↔ · Fn (𝑉 × 𝑉)))
10 fndmu 6429 . . . . . . . . . . . 12 (( · Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → (𝑉 × 𝑉) = (𝐾 × 𝑉))
1110ex 416 . . . . . . . . . . 11 ( · Fn (𝑉 × 𝑉) → ( · Fn (𝐾 × 𝑉) → (𝑉 × 𝑉) = (𝐾 × 𝑉)))
129, 11syl6bi 256 . . . . . . . . . 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 411 . . . . . . . 8 (( + Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → ( + = · → (𝑉 × 𝑉) = (𝐾 × 𝑉)))
151lmodbn0 19640 . . . . . . . . . . 11 (𝑊 ∈ LMod → 𝑉 ≠ ∅)
16 xp11 5999 . . . . . . . . . . 11 ((𝑉 ≠ ∅ ∧ 𝑉 ≠ ∅) → ((𝑉 × 𝑉) = (𝐾 × 𝑉) ↔ (𝑉 = 𝐾𝑉 = 𝑉)))
1715, 15, 16syl2anc 587 . . . . . . . . . 10 (𝑊 ∈ LMod → ((𝑉 × 𝑉) = (𝐾 × 𝑉) ↔ (𝑉 = 𝐾𝑉 = 𝑉)))
1817simprbda 502 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ (𝑉 × 𝑉) = (𝐾 × 𝑉)) → 𝑉 = 𝐾)
1918expcom 417 . . . . . . . 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 416 . . . . 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 410 1 ((𝑊 ∈ LMod ∧ + = · ) → 𝑉 = 𝐾)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∅c0 4243   × cxp 5517   Fn wfn 6319  ‘cfv 6324  Basecbs 16477  Scalarcsca 16562  +𝑓cplusf 17843  LModclmod 19630   ·sf cscaf 19631 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7673  df-2nd 7674  df-0g 16709  df-plusf 17845  df-mgm 17846  df-sgrp 17895  df-mnd 17906  df-grp 18100  df-lmod 19632  df-scaf 19633 This theorem is referenced by:  lmodfopnelem2  19667
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