Theorem lmodfopnelem1 18722

Description: Lemma 1 for lmodfopne 18724. (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 18708	. . . 4 ⊢ (𝑊 ∈ LMod → · :(𝐾 × 𝑉)⟶𝑉)
6	5	ffnd 5959	. . 3 ⊢ (𝑊 ∈ LMod → · Fn (𝐾 × 𝑉))
7		lmodfopne.a	. . . . 5 ⊢ + = (+𝑓‘𝑊)
8	1, 7	plusffn 17073	. . . 4 ⊢ + Fn (𝑉 × 𝑉)
9		fneq1 5893	. . . . . . . . . . 11 ⊢ ( + = · → ( + Fn (𝑉 × 𝑉) ↔ · Fn (𝑉 × 𝑉)))
10		fndmu 5906	. . . . . . . . . . . 12 ⊢ (( · Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → (𝑉 × 𝑉) = (𝐾 × 𝑉))
11	10	ex 449	. . . . . . . . . . 11 ⊢ ( · Fn (𝑉 × 𝑉) → ( · Fn (𝐾 × 𝑉) → (𝑉 × 𝑉) = (𝐾 × 𝑉)))
12	9, 11	syl6bi 242	. . . . . . . . . 10 ⊢ ( + = · → ( + Fn (𝑉 × 𝑉) → ( · Fn (𝐾 × 𝑉) → (𝑉 × 𝑉) = (𝐾 × 𝑉))))
13	12	com13 86	. . . . . . . . 9 ⊢ ( · Fn (𝐾 × 𝑉) → ( + Fn (𝑉 × 𝑉) → ( + = · → (𝑉 × 𝑉) = (𝐾 × 𝑉))))
14	13	impcom 445	. . . . . . . 8 ⊢ (( + Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → ( + = · → (𝑉 × 𝑉) = (𝐾 × 𝑉)))
15	1	lmodbn0 18696	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LMod → 𝑉 ≠ ∅)
16		xp11 5488	. . . . . . . . . . 11 ⊢ ((𝑉 ≠ ∅ ∧ 𝑉 ≠ ∅) → ((𝑉 × 𝑉) = (𝐾 × 𝑉) ↔ (𝑉 = 𝐾 ∧ 𝑉 = 𝑉)))
17	15, 15, 16	syl2anc 691	. . . . . . . . . 10 ⊢ (𝑊 ∈ LMod → ((𝑉 × 𝑉) = (𝐾 × 𝑉) ↔ (𝑉 = 𝐾 ∧ 𝑉 = 𝑉)))
18	17	simprbda 651	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ (𝑉 × 𝑉) = (𝐾 × 𝑉)) → 𝑉 = 𝐾)
19	18	expcom 450	. . . . . . . 8 ⊢ ((𝑉 × 𝑉) = (𝐾 × 𝑉) → (𝑊 ∈ LMod → 𝑉 = 𝐾))
20	14, 19	syl6 34	. . . . . . 7 ⊢ (( + Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → ( + = · → (𝑊 ∈ LMod → 𝑉 = 𝐾)))
21	20	com23 84	. . . . . 6 ⊢ (( + Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → (𝑊 ∈ LMod → ( + = · → 𝑉 = 𝐾)))
22	21	ex 449	. . . . 5 ⊢ ( + Fn (𝑉 × 𝑉) → ( · Fn (𝐾 × 𝑉) → (𝑊 ∈ LMod → ( + = · → 𝑉 = 𝐾))))
23	22	com23 84	. . . 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 444	1 ⊢ ((𝑊 ∈ LMod ∧ + = · ) → 𝑉 = 𝐾)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∅c0 3874   × cxp 5036   Fn wfn 5799  ‘cfv 5804  Basecbs 15695  Scalarcsca 15771  +_𝑓cplusf 17062  LModclmod 18686   ·_sf cscaf 18687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-slot 15699  df-base 15700  df-0g 15925  df-plusf 17064  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-lmod 18688  df-scaf 18689

This theorem is referenced by:  lmodfopnelem2  18723