Theorem lmodfopnelem1 19670

Description: Lemma 1 for lmodfopne 19672. (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 19656	. . . 4 ⊢ (𝑊 ∈ LMod → · :(𝐾 × 𝑉)⟶𝑉)
6	5	ffnd 6515	. . 3 ⊢ (𝑊 ∈ LMod → · Fn (𝐾 × 𝑉))
7		lmodfopne.a	. . . . 5 ⊢ + = (+𝑓‘𝑊)
8	1, 7	plusffn 17861	. . . 4 ⊢ + Fn (𝑉 × 𝑉)
9		fneq1 6444	. . . . . . . . . . 11 ⊢ ( + = · → ( + Fn (𝑉 × 𝑉) ↔ · Fn (𝑉 × 𝑉)))
10		fndmu 6458	. . . . . . . . . . . 12 ⊢ (( · Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → (𝑉 × 𝑉) = (𝐾 × 𝑉))
11	10	ex 415	. . . . . . . . . . 11 ⊢ ( · Fn (𝑉 × 𝑉) → ( · Fn (𝐾 × 𝑉) → (𝑉 × 𝑉) = (𝐾 × 𝑉)))
12	9, 11	syl6bi 255	. . . . . . . . . 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 410	. . . . . . . 8 ⊢ (( + Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → ( + = · → (𝑉 × 𝑉) = (𝐾 × 𝑉)))
15	1	lmodbn0 19644	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LMod → 𝑉 ≠ ∅)
16		xp11 6032	. . . . . . . . . . 11 ⊢ ((𝑉 ≠ ∅ ∧ 𝑉 ≠ ∅) → ((𝑉 × 𝑉) = (𝐾 × 𝑉) ↔ (𝑉 = 𝐾 ∧ 𝑉 = 𝑉)))
17	15, 15, 16	syl2anc 586	. . . . . . . . . 10 ⊢ (𝑊 ∈ LMod → ((𝑉 × 𝑉) = (𝐾 × 𝑉) ↔ (𝑉 = 𝐾 ∧ 𝑉 = 𝑉)))
18	17	simprbda 501	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ (𝑉 × 𝑉) = (𝐾 × 𝑉)) → 𝑉 = 𝐾)
19	18	expcom 416	. . . . . . . 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 415	. . . . 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 409	1 ⊢ ((𝑊 ∈ LMod ∧ + = · ) → 𝑉 = 𝐾)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ≠ wne 3016  ∅c0 4291   × cxp 5553   Fn wfn 6350  ‘cfv 6355  Basecbs 16483  Scalarcsca 16568  +_𝑓cplusf 17849  LModclmod 19634   ·_sf cscaf 19635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7689  df-2nd 7690  df-0g 16715  df-plusf 17851  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-grp 18106  df-lmod 19636  df-scaf 19637

This theorem is referenced by:  lmodfopnelem2  19671