Theorem lmodfopnelem1 21019

Description: Lemma 1 for lmodfopne 21021. (Contributed by AV, 2-Oct-2021.)

Hypotheses

Ref	Expression
lmodfopne.t	⊢ · = ( ·sf ‘𝑊)
lmodfopne.a	⊢ + = (+𝑓‘𝑊)
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

Step	Hyp	Ref	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 ⊢ · = ( ·sf ‘𝑊)
5	1, 2, 3, 4	lmodscaf 21005	. . . 4 ⊢ (𝑊 ∈ LMod → · :(𝐾 × 𝑉)⟶𝑉)
6	5	ffnd 6706	. . 3 ⊢ (𝑊 ∈ LMod → · Fn (𝐾 × 𝑉))
7		lmodfopne.a	. . . . 5 ⊢ + = (+𝑓‘𝑊)
8	1, 7	plusffn 18702	. . . 4 ⊢ + Fn (𝑉 × 𝑉)
9		fneq1 6626	. . . . . . . . . . 11 ⊢ ( + = · → ( + Fn (𝑉 × 𝑉) ↔ · Fn (𝑉 × 𝑉)))
10		fndmu 6642	. . . . . . . . . . . 12 ⊢ (( · Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → (𝑉 × 𝑉) = (𝐾 × 𝑉))
11	10	ex 417	. . . . . . . . . . 11 ⊢ ( · Fn (𝑉 × 𝑉) → ( · Fn (𝐾 × 𝑉) → (𝑉 × 𝑉) = (𝐾 × 𝑉)))
12	9, 11	biimtrdi 256	. . . . . . . . . 10 ⊢ ( + = · → ( + Fn (𝑉 × 𝑉) → ( · Fn (𝐾 × 𝑉) → (𝑉 × 𝑉) = (𝐾 × 𝑉))))
13	12	com13 89	. . . . . . . . 9 ⊢ ( · Fn (𝐾 × 𝑉) → ( + Fn (𝑉 × 𝑉) → ( + = · → (𝑉 × 𝑉) = (𝐾 × 𝑉))))
14	13	impcom 412	. . . . . . . 8 ⊢ (( + Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → ( + = · → (𝑉 × 𝑉) = (𝐾 × 𝑉)))
15	1	lmodbn0 20992	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LMod → 𝑉 ≠ ∅)
16		xp11 6173	. . . . . . . . . . 11 ⊢ ((𝑉 ≠ ∅ ∧ 𝑉 ≠ ∅) → ((𝑉 × 𝑉) = (𝐾 × 𝑉) ↔ (𝑉 = 𝐾 ∧ 𝑉 = 𝑉)))
17	15, 15, 16	syl2anc 595	. . . . . . . . . 10 ⊢ (𝑊 ∈ LMod → ((𝑉 × 𝑉) = (𝐾 × 𝑉) ↔ (𝑉 = 𝐾 ∧ 𝑉 = 𝑉)))
18	17	simprbda 503	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ (𝑉 × 𝑉) = (𝐾 × 𝑉)) → 𝑉 = 𝐾)
19	18	expcom 418	. . . . . . . 8 ⊢ ((𝑉 × 𝑉) = (𝐾 × 𝑉) → (𝑊 ∈ LMod → 𝑉 = 𝐾))
20	14, 19	syl6 36	. . . . . . 7 ⊢ (( + Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → ( + = · → (𝑊 ∈ LMod → 𝑉 = 𝐾)))
21	20	com23 87	. . . . . 6 ⊢ (( + Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → (𝑊 ∈ LMod → ( + = · → 𝑉 = 𝐾)))
22	21	ex 417	. . . . 5 ⊢ ( + Fn (𝑉 × 𝑉) → ( · Fn (𝐾 × 𝑉) → (𝑊 ∈ LMod → ( + = · → 𝑉 = 𝐾))))
23	22	com23 87	. . . 4 ⊢ ( + Fn (𝑉 × 𝑉) → (𝑊 ∈ LMod → ( · Fn (𝐾 × 𝑉) → ( + = · → 𝑉 = 𝐾))))
24	8, 23	ax-mp 5	. . 3 ⊢ (𝑊 ∈ LMod → ( · Fn (𝐾 × 𝑉) → ( + = · → 𝑉 = 𝐾)))
25	6, 24	mpd 16	. 2 ⊢ (𝑊 ∈ LMod → ( + = · → 𝑉 = 𝐾))
26	25	imp 411	1 ⊢ ((𝑊 ∈ LMod ∧ + = · ) → 𝑉 = 𝐾)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∅c0 4286   × cxp 5659   Fn wfn 6531  ‘cfv 6536  Basecbs 17264  Scalarcsca 17308  +_𝑓cplusf 18690  LModclmod 20981   ·_sf cscaf 20982

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-0g 17489  df-plusf 18692  df-mgm 18693  df-sgrp 18772  df-mnd 18788  df-grp 18998  df-lmod 20983  df-scaf 20984

This theorem is referenced by:  lmodfopnelem2  21020