Theorem lmodfopnelem1 14403

Description: Lemma 1 for lmodfopne 14405. (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

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lmodfopne.v	. . . 4 ⊢ 𝑉 = (Base‘𝑊)
2		lmodfopne.a	. . . 4 ⊢ + = (+𝑓‘𝑊)
3	1, 2	plusffng 13511	. . 3 ⊢ (𝑊 ∈ LMod → + Fn (𝑉 × 𝑉))
4		lmodfopne.s	. . . 4 ⊢ 𝑆 = (Scalar‘𝑊)
5		lmodfopne.k	. . . 4 ⊢ 𝐾 = (Base‘𝑆)
6		lmodfopne.t	. . . 4 ⊢ · = ( ·sf ‘𝑊)
7	1, 4, 5, 6	scaffng 14388	. . 3 ⊢ (𝑊 ∈ LMod → · Fn (𝐾 × 𝑉))
8		fneq1 5425	. . . . . . . . . 10 ⊢ ( + = · → ( + Fn (𝑉 × 𝑉) ↔ · Fn (𝑉 × 𝑉)))
9		fndmu 5440	. . . . . . . . . . 11 ⊢ (( · Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → (𝑉 × 𝑉) = (𝐾 × 𝑉))
10	9	ex 115	. . . . . . . . . 10 ⊢ ( · Fn (𝑉 × 𝑉) → ( · Fn (𝐾 × 𝑉) → (𝑉 × 𝑉) = (𝐾 × 𝑉)))
11	8, 10	biimtrdi 163	. . . . . . . . 9 ⊢ ( + = · → ( + Fn (𝑉 × 𝑉) → ( · Fn (𝐾 × 𝑉) → (𝑉 × 𝑉) = (𝐾 × 𝑉))))
12	11	com13 80	. . . . . . . 8 ⊢ ( · Fn (𝐾 × 𝑉) → ( + Fn (𝑉 × 𝑉) → ( + = · → (𝑉 × 𝑉) = (𝐾 × 𝑉))))
13	12	impcom 125	. . . . . . 7 ⊢ (( + Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → ( + = · → (𝑉 × 𝑉) = (𝐾 × 𝑉)))
14		lmodgrp 14373	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
15		eqid 2231	. . . . . . . . . . . 12 ⊢ (0g‘𝑊) = (0g‘𝑊)
16	1, 15	grpidcl 13675	. . . . . . . . . . 11 ⊢ (𝑊 ∈ Grp → (0g‘𝑊) ∈ 𝑉)
17		elex2 2820	. . . . . . . . . . 11 ⊢ ((0g‘𝑊) ∈ 𝑉 → ∃𝑤 𝑤 ∈ 𝑉)
18	14, 16, 17	3syl 17	. . . . . . . . . 10 ⊢ (𝑊 ∈ LMod → ∃𝑤 𝑤 ∈ 𝑉)
19		xp11m 5182	. . . . . . . . . 10 ⊢ ((∃𝑤 𝑤 ∈ 𝑉 ∧ ∃𝑤 𝑤 ∈ 𝑉) → ((𝑉 × 𝑉) = (𝐾 × 𝑉) ↔ (𝑉 = 𝐾 ∧ 𝑉 = 𝑉)))
20	18, 18, 19	syl2anc 411	. . . . . . . . 9 ⊢ (𝑊 ∈ LMod → ((𝑉 × 𝑉) = (𝐾 × 𝑉) ↔ (𝑉 = 𝐾 ∧ 𝑉 = 𝑉)))
21	20	simprbda 383	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝑉 × 𝑉) = (𝐾 × 𝑉)) → 𝑉 = 𝐾)
22	21	expcom 116	. . . . . . 7 ⊢ ((𝑉 × 𝑉) = (𝐾 × 𝑉) → (𝑊 ∈ LMod → 𝑉 = 𝐾))
23	13, 22	syl6 33	. . . . . 6 ⊢ (( + Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → ( + = · → (𝑊 ∈ LMod → 𝑉 = 𝐾)))
24	23	com23 78	. . . . 5 ⊢ (( + Fn (𝑉 × 𝑉) ∧ · Fn (𝐾 × 𝑉)) → (𝑊 ∈ LMod → ( + = · → 𝑉 = 𝐾)))
25	24	ex 115	. . . 4 ⊢ ( + Fn (𝑉 × 𝑉) → ( · Fn (𝐾 × 𝑉) → (𝑊 ∈ LMod → ( + = · → 𝑉 = 𝐾))))
26	25	com3r 79	. . 3 ⊢ (𝑊 ∈ LMod → ( + Fn (𝑉 × 𝑉) → ( · Fn (𝐾 × 𝑉) → ( + = · → 𝑉 = 𝐾))))
27	3, 7, 26	mp2d 47	. 2 ⊢ (𝑊 ∈ LMod → ( + = · → 𝑉 = 𝐾))
28	27	imp 124	1 ⊢ ((𝑊 ∈ LMod ∧ + = · ) → 𝑉 = 𝐾)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398  ∃wex 1541   ∈ wcel 2202   × cxp 4729   Fn wfn 5328  ‘cfv 5333  Basecbs 13145  Scalarcsca 13226  0_gc0g 13402  +_𝑓cplusf 13499  Grpcgrp 13646  LModclmod 14366   ·_sf cscaf 14367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-5 9247  df-6 9248  df-ndx 13148  df-slot 13149  df-base 13151  df-plusg 13236  df-mulr 13237  df-sca 13239  df-vsca 13240  df-0g 13404  df-plusf 13501  df-mgm 13502  df-sgrp 13548  df-mnd 13563  df-grp 13649  df-lmod 14368  df-scaf 14369

This theorem is referenced by:  lmodfopnelem2  14404