Theorem lmodfopnelem1 20888

Description: Lemma 1 for lmodfopne 20890. (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 20874	. . . 4 ⊢ (𝑊 ∈ LMod → · :(𝐾 × 𝑉)⟶𝑉)
6	5	ffnd 6656	. . 3 ⊢ (𝑊 ∈ LMod → · Fn (𝐾 × 𝑉))
7		lmodfopne.a	. . . . 5 ⊢ + = (+𝑓‘𝑊)
8	1, 7	plusffn 18608	. . . 4 ⊢ + Fn (𝑉 × 𝑉)
9		fneq1 6576	. . . . . . . . . . 11 ⊢ ( + = · → ( + Fn (𝑉 × 𝑉) ↔ · Fn (𝑉 × 𝑉)))
10		fndmu 6592	. . . . . . . . . . . 12 ⊢ (( · Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → (𝑉 × 𝑉) = (𝐾 × 𝑉))
11	10	ex 413	. . . . . . . . . . 11 ⊢ ( · Fn (𝑉 × 𝑉) → ( · Fn (𝐾 × 𝑉) → (𝑉 × 𝑉) = (𝐾 × 𝑉)))
12	9, 11	biimtrdi 254	. . . . . . . . . 10 ⊢ ( + = · → ( + Fn (𝑉 × 𝑉) → ( · Fn (𝐾 × 𝑉) → (𝑉 × 𝑉) = (𝐾 × 𝑉))))
13	12	com13 88	. . . . . . . . 9 ⊢ ( · Fn (𝐾 × 𝑉) → ( + Fn (𝑉 × 𝑉) → ( + = · → (𝑉 × 𝑉) = (𝐾 × 𝑉))))
14	13	impcom 408	. . . . . . . 8 ⊢ (( + Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → ( + = · → (𝑉 × 𝑉) = (𝐾 × 𝑉)))
15	1	lmodbn0 20861	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LMod → 𝑉 ≠ ∅)
16		xp11 6126	. . . . . . . . . . 11 ⊢ ((𝑉 ≠ ∅ ∧ 𝑉 ≠ ∅) → ((𝑉 × 𝑉) = (𝐾 × 𝑉) ↔ (𝑉 = 𝐾 ∧ 𝑉 = 𝑉)))
17	15, 15, 16	syl2anc 590	. . . . . . . . . 10 ⊢ (𝑊 ∈ LMod → ((𝑉 × 𝑉) = (𝐾 × 𝑉) ↔ (𝑉 = 𝐾 ∧ 𝑉 = 𝑉)))
18	17	simprbda 499	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ (𝑉 × 𝑉) = (𝐾 × 𝑉)) → 𝑉 = 𝐾)
19	18	expcom 414	. . . . . . . 8 ⊢ ((𝑉 × 𝑉) = (𝐾 × 𝑉) → (𝑊 ∈ LMod → 𝑉 = 𝐾))
20	14, 19	syl6 35	. . . . . . 7 ⊢ (( + Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → ( + = · → (𝑊 ∈ LMod → 𝑉 = 𝐾)))
21	20	com23 86	. . . . . 6 ⊢ (( + Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → (𝑊 ∈ LMod → ( + = · → 𝑉 = 𝐾)))
22	21	ex 413	. . . . 5 ⊢ ( + Fn (𝑉 × 𝑉) → ( · Fn (𝐾 × 𝑉) → (𝑊 ∈ LMod → ( + = · → 𝑉 = 𝐾))))
23	22	com23 86	. . . 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 15	. 2 ⊢ (𝑊 ∈ LMod → ( + = · → 𝑉 = 𝐾))
26	25	imp 407	1 ⊢ ((𝑊 ∈ LMod ∧ + = · ) → 𝑉 = 𝐾)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ∅c0 4261   × cxp 5616   Fn wfn 6480  ‘cfv 6485  Basecbs 17170  Scalarcsca 17214  +_𝑓cplusf 18596  LModclmod 20850   ·_sf cscaf 20851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-0g 17395  df-plusf 18598  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-grp 18903  df-lmod 20852  df-scaf 20853

This theorem is referenced by:  lmodfopnelem2  20889