Theorem lmodfopnelem1 20849

Description: Lemma 1 for lmodfopne 20851. (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 20835	. . . 4 ⊢ (𝑊 ∈ LMod → · :(𝐾 × 𝑉)⟶𝑉)
6	5	ffnd 6663	. . 3 ⊢ (𝑊 ∈ LMod → · Fn (𝐾 × 𝑉))
7		lmodfopne.a	. . . . 5 ⊢ + = (+𝑓‘𝑊)
8	1, 7	plusffn 18574	. . . 4 ⊢ + Fn (𝑉 × 𝑉)
9		fneq1 6583	. . . . . . . . . . 11 ⊢ ( + = · → ( + Fn (𝑉 × 𝑉) ↔ · Fn (𝑉 × 𝑉)))
10		fndmu 6599	. . . . . . . . . . . 12 ⊢ (( · Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → (𝑉 × 𝑉) = (𝐾 × 𝑉))
11	10	ex 412	. . . . . . . . . . 11 ⊢ ( · Fn (𝑉 × 𝑉) → ( · Fn (𝐾 × 𝑉) → (𝑉 × 𝑉) = (𝐾 × 𝑉)))
12	9, 11	biimtrdi 253	. . . . . . . . . 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 407	. . . . . . . 8 ⊢ (( + Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → ( + = · → (𝑉 × 𝑉) = (𝐾 × 𝑉)))
15	1	lmodbn0 20822	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LMod → 𝑉 ≠ ∅)
16		xp11 6133	. . . . . . . . . . 11 ⊢ ((𝑉 ≠ ∅ ∧ 𝑉 ≠ ∅) → ((𝑉 × 𝑉) = (𝐾 × 𝑉) ↔ (𝑉 = 𝐾 ∧ 𝑉 = 𝑉)))
17	15, 15, 16	syl2anc 584	. . . . . . . . . 10 ⊢ (𝑊 ∈ LMod → ((𝑉 × 𝑉) = (𝐾 × 𝑉) ↔ (𝑉 = 𝐾 ∧ 𝑉 = 𝑉)))
18	17	simprbda 498	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ (𝑉 × 𝑉) = (𝐾 × 𝑉)) → 𝑉 = 𝐾)
19	18	expcom 413	. . . . . . . 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 412	. . . . 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 406	1 ⊢ ((𝑊 ∈ LMod ∧ + = · ) → 𝑉 = 𝐾)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∅c0 4285   × cxp 5622   Fn wfn 6487  ‘cfv 6492  Basecbs 17136  Scalarcsca 17180  +_𝑓cplusf 18562  LModclmod 20811   ·_sf cscaf 20812

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-0g 17361  df-plusf 18564  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18866  df-lmod 20813  df-scaf 20814

This theorem is referenced by:  lmodfopnelem2  20850