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Theorem hdmapglem7 36037
Description: Lemma for hdmapg 36038. Line 15 in [Baer] p. 111, f(x,y) alpha = f(y,x). In the proof, our 𝐸, (𝑂‘{𝐸}) 𝑋, 𝑌, 𝑘, 𝑢, 𝑙, 𝑣 correspond to Baer's w, H, x, y, x', x'', y' , y'', and our ((𝑆𝑌)‘𝑋) corresponds to Baer's f(x,y). (Contributed by NM, 14-Jun-2015.)
Hypotheses
Ref Expression
hdmapglem7.h 𝐻 = (LHyp‘𝐾)
hdmapglem7.e 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩
hdmapglem7.o 𝑂 = ((ocH‘𝐾)‘𝑊)
hdmapglem7.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmapglem7.v 𝑉 = (Base‘𝑈)
hdmapglem7.p + = (+g𝑈)
hdmapglem7.q · = ( ·𝑠𝑈)
hdmapglem7.r 𝑅 = (Scalar‘𝑈)
hdmapglem7.b 𝐵 = (Base‘𝑅)
hdmapglem7.a = (LSSum‘𝑈)
hdmapglem7.n 𝑁 = (LSpan‘𝑈)
hdmapglem7.k (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
hdmapglem7.x (𝜑𝑋𝑉)
hdmapglem7.t × = (.r𝑅)
hdmapglem7.z 0 = (0g𝑅)
hdmapglem7.c = (+g𝑅)
hdmapglem7.s 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmapglem7.g 𝐺 = ((HGMap‘𝐾)‘𝑊)
hdmapglem7.y (𝜑𝑌𝑉)
Assertion
Ref Expression
hdmapglem7 (𝜑 → (𝐺‘((𝑆𝑌)‘𝑋)) = ((𝑆𝑋)‘𝑌))

Proof of Theorem hdmapglem7
Dummy variables 𝑘 𝑙 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hdmapglem7.h . . 3 𝐻 = (LHyp‘𝐾)
2 hdmapglem7.e . . 3 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩
3 hdmapglem7.o . . 3 𝑂 = ((ocH‘𝐾)‘𝑊)
4 hdmapglem7.u . . 3 𝑈 = ((DVecH‘𝐾)‘𝑊)
5 hdmapglem7.v . . 3 𝑉 = (Base‘𝑈)
6 hdmapglem7.p . . 3 + = (+g𝑈)
7 hdmapglem7.q . . 3 · = ( ·𝑠𝑈)
8 hdmapglem7.r . . 3 𝑅 = (Scalar‘𝑈)
9 hdmapglem7.b . . 3 𝐵 = (Base‘𝑅)
10 hdmapglem7.a . . 3 = (LSSum‘𝑈)
11 hdmapglem7.n . . 3 𝑁 = (LSpan‘𝑈)
12 hdmapglem7.k . . 3 (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
13 hdmapglem7.x . . 3 (𝜑𝑋𝑉)
141, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13hdmapglem7a 36035 . 2 (𝜑 → ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑘𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢))
15 hdmapglem7.y . . 3 (𝜑𝑌𝑉)
161, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15hdmapglem7a 36035 . 2 (𝜑 → ∃𝑣 ∈ (𝑂‘{𝐸})∃𝑙𝐵 𝑌 = ((𝑙 · 𝐸) + 𝑣))
17 hdmapglem7.c . . . . . . . . . . . 12 = (+g𝑅)
18 hdmapglem7.g . . . . . . . . . . . 12 𝐺 = ((HGMap‘𝐾)‘𝑊)
1912ad2antrr 757 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
201, 4, 12dvhlmod 35215 . . . . . . . . . . . . . . 15 (𝜑𝑈 ∈ LMod)
218lmodring 18635 . . . . . . . . . . . . . . 15 (𝑈 ∈ LMod → 𝑅 ∈ Ring)
2220, 21syl 17 . . . . . . . . . . . . . 14 (𝜑𝑅 ∈ Ring)
2322ad2antrr 757 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → 𝑅 ∈ Ring)
24 simplrr 796 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → 𝑘𝐵)
25 simprr 791 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → 𝑙𝐵)
261, 4, 8, 9, 18, 19, 25hgmapcl 35997 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐺𝑙) ∈ 𝐵)
27 hdmapglem7.t . . . . . . . . . . . . . 14 × = (.r𝑅)
289, 27ringcl 18325 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑘𝐵 ∧ (𝐺𝑙) ∈ 𝐵) → (𝑘 × (𝐺𝑙)) ∈ 𝐵)
2923, 24, 26, 28syl3anc 1317 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝑘 × (𝐺𝑙)) ∈ 𝐵)
30 hdmapglem7.s . . . . . . . . . . . . 13 𝑆 = ((HDMap‘𝐾)‘𝑊)
31 eqid 2604 . . . . . . . . . . . . . . . . . . 19 (Base‘𝐾) = (Base‘𝐾)
32 eqid 2604 . . . . . . . . . . . . . . . . . . 19 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
33 eqid 2604 . . . . . . . . . . . . . . . . . . 19 (0g𝑈) = (0g𝑈)
341, 31, 32, 4, 5, 33, 2, 12dvheveccl 35217 . . . . . . . . . . . . . . . . . 18 (𝜑𝐸 ∈ (𝑉 ∖ {(0g𝑈)}))
3534eldifad 3546 . . . . . . . . . . . . . . . . 17 (𝜑𝐸𝑉)
3635snssd 4275 . . . . . . . . . . . . . . . 16 (𝜑 → {𝐸} ⊆ 𝑉)
371, 4, 5, 3dochssv 35460 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ {𝐸} ⊆ 𝑉) → (𝑂‘{𝐸}) ⊆ 𝑉)
3812, 36, 37syl2anc 690 . . . . . . . . . . . . . . 15 (𝜑 → (𝑂‘{𝐸}) ⊆ 𝑉)
3938ad2antrr 757 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝑂‘{𝐸}) ⊆ 𝑉)
40 simplrl 795 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → 𝑢 ∈ (𝑂‘{𝐸}))
4139, 40sseldd 3563 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → 𝑢𝑉)
42 simprl 789 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → 𝑣 ∈ (𝑂‘{𝐸}))
4339, 42sseldd 3563 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → 𝑣𝑉)
441, 4, 5, 8, 9, 30, 19, 41, 43hdmapipcl 36013 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → ((𝑆𝑣)‘𝑢) ∈ 𝐵)
451, 4, 8, 9, 17, 18, 19, 29, 44hgmapadd 36002 . . . . . . . . . . 11 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐺‘((𝑘 × (𝐺𝑙)) ((𝑆𝑣)‘𝑢))) = ((𝐺‘(𝑘 × (𝐺𝑙))) (𝐺‘((𝑆𝑣)‘𝑢))))
461, 4, 8, 9, 27, 18, 19, 24, 26hgmapmul 36003 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐺‘(𝑘 × (𝐺𝑙))) = ((𝐺‘(𝐺𝑙)) × (𝐺𝑘)))
471, 4, 8, 9, 18, 19, 25hgmapvv 36034 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐺‘(𝐺𝑙)) = 𝑙)
4847oveq1d 6537 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → ((𝐺‘(𝐺𝑙)) × (𝐺𝑘)) = (𝑙 × (𝐺𝑘)))
4946, 48eqtrd 2638 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐺‘(𝑘 × (𝐺𝑙))) = (𝑙 × (𝐺𝑘)))
50 eqid 2604 . . . . . . . . . . . . 13 (-g𝑈) = (-g𝑈)
51 hdmapglem7.z . . . . . . . . . . . . 13 0 = (0g𝑅)
521, 2, 3, 4, 5, 6, 50, 7, 8, 9, 27, 51, 30, 18, 19, 40, 42, 24, 24hdmapglem5 36030 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐺‘((𝑆𝑣)‘𝑢)) = ((𝑆𝑢)‘𝑣))
5349, 52oveq12d 6540 . . . . . . . . . . 11 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → ((𝐺‘(𝑘 × (𝐺𝑙))) (𝐺‘((𝑆𝑣)‘𝑢))) = ((𝑙 × (𝐺𝑘)) ((𝑆𝑢)‘𝑣)))
5445, 53eqtrd 2638 . . . . . . . . . 10 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐺‘((𝑘 × (𝐺𝑙)) ((𝑆𝑣)‘𝑢))) = ((𝑙 × (𝐺𝑘)) ((𝑆𝑢)‘𝑣)))
5513ad2antrr 757 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → 𝑋𝑉)
561, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 19, 55, 27, 51, 17, 30, 18, 42, 40, 25, 24hdmapglem7b 36036 . . . . . . . . . . 11 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → ((𝑆‘((𝑙 · 𝐸) + 𝑣))‘((𝑘 · 𝐸) + 𝑢)) = ((𝑘 × (𝐺𝑙)) ((𝑆𝑣)‘𝑢)))
5756fveq2d 6087 . . . . . . . . . 10 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐺‘((𝑆‘((𝑙 · 𝐸) + 𝑣))‘((𝑘 · 𝐸) + 𝑢))) = (𝐺‘((𝑘 × (𝐺𝑙)) ((𝑆𝑣)‘𝑢))))
581, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 19, 55, 27, 51, 17, 30, 18, 40, 42, 24, 25hdmapglem7b 36036 . . . . . . . . . 10 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → ((𝑆‘((𝑘 · 𝐸) + 𝑢))‘((𝑙 · 𝐸) + 𝑣)) = ((𝑙 × (𝐺𝑘)) ((𝑆𝑢)‘𝑣)))
5954, 57, 583eqtr4d 2648 . . . . . . . . 9 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐺‘((𝑆‘((𝑙 · 𝐸) + 𝑣))‘((𝑘 · 𝐸) + 𝑢))) = ((𝑆‘((𝑘 · 𝐸) + 𝑢))‘((𝑙 · 𝐸) + 𝑣)))
60593adantl3 1211 . . . . . . . 8 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐺‘((𝑆‘((𝑙 · 𝐸) + 𝑣))‘((𝑘 · 𝐸) + 𝑢))) = ((𝑆‘((𝑘 · 𝐸) + 𝑢))‘((𝑙 · 𝐸) + 𝑣)))
61603adant3 1073 . . . . . . 7 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → (𝐺‘((𝑆‘((𝑙 · 𝐸) + 𝑣))‘((𝑘 · 𝐸) + 𝑢))) = ((𝑆‘((𝑘 · 𝐸) + 𝑢))‘((𝑙 · 𝐸) + 𝑣)))
62 simp3 1055 . . . . . . . . . 10 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → 𝑌 = ((𝑙 · 𝐸) + 𝑣))
6362fveq2d 6087 . . . . . . . . 9 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → (𝑆𝑌) = (𝑆‘((𝑙 · 𝐸) + 𝑣)))
64 simp13 1085 . . . . . . . . 9 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → 𝑋 = ((𝑘 · 𝐸) + 𝑢))
6563, 64fveq12d 6089 . . . . . . . 8 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → ((𝑆𝑌)‘𝑋) = ((𝑆‘((𝑙 · 𝐸) + 𝑣))‘((𝑘 · 𝐸) + 𝑢)))
6665fveq2d 6087 . . . . . . 7 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → (𝐺‘((𝑆𝑌)‘𝑋)) = (𝐺‘((𝑆‘((𝑙 · 𝐸) + 𝑣))‘((𝑘 · 𝐸) + 𝑢))))
6764fveq2d 6087 . . . . . . . 8 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → (𝑆𝑋) = (𝑆‘((𝑘 · 𝐸) + 𝑢)))
6867, 62fveq12d 6089 . . . . . . 7 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → ((𝑆𝑋)‘𝑌) = ((𝑆‘((𝑘 · 𝐸) + 𝑢))‘((𝑙 · 𝐸) + 𝑣)))
6961, 66, 683eqtr4d 2648 . . . . . 6 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → (𝐺‘((𝑆𝑌)‘𝑋)) = ((𝑆𝑋)‘𝑌))
70693exp 1255 . . . . 5 ((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) → ((𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵) → (𝑌 = ((𝑙 · 𝐸) + 𝑣) → (𝐺‘((𝑆𝑌)‘𝑋)) = ((𝑆𝑋)‘𝑌))))
7170rexlimdvv 3013 . . . 4 ((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) → (∃𝑣 ∈ (𝑂‘{𝐸})∃𝑙𝐵 𝑌 = ((𝑙 · 𝐸) + 𝑣) → (𝐺‘((𝑆𝑌)‘𝑋)) = ((𝑆𝑋)‘𝑌)))
72713exp 1255 . . 3 (𝜑 → ((𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) → (𝑋 = ((𝑘 · 𝐸) + 𝑢) → (∃𝑣 ∈ (𝑂‘{𝐸})∃𝑙𝐵 𝑌 = ((𝑙 · 𝐸) + 𝑣) → (𝐺‘((𝑆𝑌)‘𝑋)) = ((𝑆𝑋)‘𝑌)))))
7372rexlimdvv 3013 . 2 (𝜑 → (∃𝑢 ∈ (𝑂‘{𝐸})∃𝑘𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢) → (∃𝑣 ∈ (𝑂‘{𝐸})∃𝑙𝐵 𝑌 = ((𝑙 · 𝐸) + 𝑣) → (𝐺‘((𝑆𝑌)‘𝑋)) = ((𝑆𝑋)‘𝑌))))
7414, 16, 73mp2d 46 1 (𝜑 → (𝐺‘((𝑆𝑌)‘𝑋)) = ((𝑆𝑋)‘𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1975  wrex 2891  wss 3534  {csn 4119  cop 4125   I cid 4933  cres 5025  cfv 5785  (class class class)co 6522  Basecbs 15636  +gcplusg 15709  .rcmulr 15710  Scalarcsca 15712   ·𝑠 cvsca 15713  0gc0g 15864  -gcsg 17188  LSSumclsm 17813  Ringcrg 18311  LModclmod 18627  LSpanclspn 18733  HLchlt 33453  LHypclh 34086  LTrncltrn 34203  DVecHcdvh 35183  ocHcoch 35452  HDMapchdma 35898  HGMapchg 35991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819  ax-cnex 9843  ax-resscn 9844  ax-1cn 9845  ax-icn 9846  ax-addcl 9847  ax-addrcl 9848  ax-mulcl 9849  ax-mulrcl 9850  ax-mulcom 9851  ax-addass 9852  ax-mulass 9853  ax-distr 9854  ax-i2m1 9855  ax-1ne0 9856  ax-1rid 9857  ax-rnegex 9858  ax-rrecex 9859  ax-cnre 9860  ax-pre-lttri 9861  ax-pre-lttrn 9862  ax-pre-ltadd 9863  ax-pre-mulgt0 9864  ax-riotaBAD 33055
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-nel 2777  df-ral 2895  df-rex 2896  df-reu 2897  df-rmo 2898  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-pss 3550  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-tp 4124  df-op 4126  df-ot 4128  df-uni 4362  df-int 4400  df-iun 4446  df-iin 4447  df-br 4573  df-opab 4633  df-mpt 4634  df-tr 4670  df-eprel 4934  df-id 4938  df-po 4944  df-so 4945  df-fr 4982  df-we 4984  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-pred 5578  df-ord 5624  df-on 5625  df-lim 5626  df-suc 5627  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-riota 6484  df-ov 6525  df-oprab 6526  df-mpt2 6527  df-of 6767  df-om 6930  df-1st 7031  df-2nd 7032  df-tpos 7211  df-undef 7258  df-wrecs 7266  df-recs 7327  df-rdg 7365  df-1o 7419  df-oadd 7423  df-er 7601  df-map 7718  df-en 7814  df-dom 7815  df-sdom 7816  df-fin 7817  df-pnf 9927  df-mnf 9928  df-xr 9929  df-ltxr 9930  df-le 9931  df-sub 10114  df-neg 10115  df-nn 10863  df-2 10921  df-3 10922  df-4 10923  df-5 10924  df-6 10925  df-n0 11135  df-z 11206  df-uz 11515  df-fz 12148  df-struct 15638  df-ndx 15639  df-slot 15640  df-base 15641  df-sets 15642  df-ress 15643  df-plusg 15722  df-mulr 15723  df-sca 15725  df-vsca 15726  df-0g 15866  df-mre 16010  df-mrc 16011  df-acs 16013  df-preset 16692  df-poset 16710  df-plt 16722  df-lub 16738  df-glb 16739  df-join 16740  df-meet 16741  df-p0 16803  df-p1 16804  df-lat 16810  df-clat 16872  df-mgm 17006  df-sgrp 17048  df-mnd 17059  df-submnd 17100  df-grp 17189  df-minusg 17190  df-sbg 17191  df-subg 17355  df-cntz 17514  df-oppg 17540  df-lsm 17815  df-cmn 17959  df-abl 17960  df-mgp 18254  df-ur 18266  df-ring 18313  df-oppr 18387  df-dvdsr 18405  df-unit 18406  df-invr 18436  df-dvr 18447  df-drng 18513  df-lmod 18629  df-lss 18695  df-lsp 18734  df-lvec 18865  df-lsatoms 33079  df-lshyp 33080  df-lcv 33122  df-lfl 33161  df-lkr 33189  df-ldual 33227  df-oposet 33279  df-ol 33281  df-oml 33282  df-covers 33369  df-ats 33370  df-atl 33401  df-cvlat 33425  df-hlat 33454  df-llines 33600  df-lplanes 33601  df-lvols 33602  df-lines 33603  df-psubsp 33605  df-pmap 33606  df-padd 33898  df-lhyp 34090  df-laut 34091  df-ldil 34206  df-ltrn 34207  df-trl 34262  df-tgrp 34847  df-tendo 34859  df-edring 34861  df-dveca 35107  df-disoa 35134  df-dvech 35184  df-dib 35244  df-dic 35278  df-dih 35334  df-doch 35453  df-djh 35500  df-lcdual 35692  df-mapd 35730  df-hvmap 35862  df-hdmap1 35899  df-hdmap 35900  df-hgmap 35992
This theorem is referenced by:  hdmapg  36038
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