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Theorem hdmapglem7 39170
Description: Lemma for hdmapg 39171. 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 39168 . 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 39168 . 2 (𝜑 → ∃𝑣 ∈ (𝑂‘{𝐸})∃𝑙𝐵 𝑌 = ((𝑙 · 𝐸) + 𝑣))
17 hdmapglem7.c . . . . . . . . . . . 12 = (+g𝑅)
18 hdmapglem7.g . . . . . . . . . . . 12 𝐺 = ((HGMap‘𝐾)‘𝑊)
1912ad2antrr 725 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
201, 4, 12dvhlmod 38351 . . . . . . . . . . . . . . 15 (𝜑𝑈 ∈ LMod)
218lmodring 19642 . . . . . . . . . . . . . . 15 (𝑈 ∈ LMod → 𝑅 ∈ Ring)
2220, 21syl 17 . . . . . . . . . . . . . 14 (𝜑𝑅 ∈ Ring)
2322ad2antrr 725 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → 𝑅 ∈ Ring)
24 simplrr 777 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → 𝑘𝐵)
25 simprr 772 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → 𝑙𝐵)
261, 4, 8, 9, 18, 19, 25hgmapcl 39130 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐺𝑙) ∈ 𝐵)
27 hdmapglem7.t . . . . . . . . . . . . . 14 × = (.r𝑅)
289, 27ringcl 19314 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑘𝐵 ∧ (𝐺𝑙) ∈ 𝐵) → (𝑘 × (𝐺𝑙)) ∈ 𝐵)
2923, 24, 26, 28syl3anc 1368 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝑘 × (𝐺𝑙)) ∈ 𝐵)
30 hdmapglem7.s . . . . . . . . . . . . 13 𝑆 = ((HDMap‘𝐾)‘𝑊)
31 eqid 2824 . . . . . . . . . . . . . . . . . . 19 (Base‘𝐾) = (Base‘𝐾)
32 eqid 2824 . . . . . . . . . . . . . . . . . . 19 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
33 eqid 2824 . . . . . . . . . . . . . . . . . . 19 (0g𝑈) = (0g𝑈)
341, 31, 32, 4, 5, 33, 2, 12dvheveccl 38353 . . . . . . . . . . . . . . . . . 18 (𝜑𝐸 ∈ (𝑉 ∖ {(0g𝑈)}))
3534eldifad 3931 . . . . . . . . . . . . . . . . 17 (𝜑𝐸𝑉)
3635snssd 4726 . . . . . . . . . . . . . . . 16 (𝜑 → {𝐸} ⊆ 𝑉)
371, 4, 5, 3dochssv 38596 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ {𝐸} ⊆ 𝑉) → (𝑂‘{𝐸}) ⊆ 𝑉)
3812, 36, 37syl2anc 587 . . . . . . . . . . . . . . 15 (𝜑 → (𝑂‘{𝐸}) ⊆ 𝑉)
3938ad2antrr 725 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝑂‘{𝐸}) ⊆ 𝑉)
40 simplrl 776 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → 𝑢 ∈ (𝑂‘{𝐸}))
4139, 40sseldd 3954 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → 𝑢𝑉)
42 simprl 770 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → 𝑣 ∈ (𝑂‘{𝐸}))
4339, 42sseldd 3954 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → 𝑣𝑉)
441, 4, 5, 8, 9, 30, 19, 41, 43hdmapipcl 39146 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → ((𝑆𝑣)‘𝑢) ∈ 𝐵)
451, 4, 8, 9, 17, 18, 19, 29, 44hgmapadd 39135 . . . . . . . . . . 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 39136 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐺‘(𝑘 × (𝐺𝑙))) = ((𝐺‘(𝐺𝑙)) × (𝐺𝑘)))
471, 4, 8, 9, 18, 19, 25hgmapvv 39167 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐺‘(𝐺𝑙)) = 𝑙)
4847oveq1d 7164 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → ((𝐺‘(𝐺𝑙)) × (𝐺𝑘)) = (𝑙 × (𝐺𝑘)))
4946, 48eqtrd 2859 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐺‘(𝑘 × (𝐺𝑙))) = (𝑙 × (𝐺𝑘)))
50 eqid 2824 . . . . . . . . . . . . 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 39163 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐺‘((𝑆𝑣)‘𝑢)) = ((𝑆𝑢)‘𝑣))
5349, 52oveq12d 7167 . . . . . . . . . . 11 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → ((𝐺‘(𝑘 × (𝐺𝑙))) (𝐺‘((𝑆𝑣)‘𝑢))) = ((𝑙 × (𝐺𝑘)) ((𝑆𝑢)‘𝑣)))
5445, 53eqtrd 2859 . . . . . . . . . 10 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐺‘((𝑘 × (𝐺𝑙)) ((𝑆𝑣)‘𝑢))) = ((𝑙 × (𝐺𝑘)) ((𝑆𝑢)‘𝑣)))
5513ad2antrr 725 . . . . . . . . . . . 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 39169 . . . . . . . . . . 11 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → ((𝑆‘((𝑙 · 𝐸) + 𝑣))‘((𝑘 · 𝐸) + 𝑢)) = ((𝑘 × (𝐺𝑙)) ((𝑆𝑣)‘𝑢)))
5756fveq2d 6665 . . . . . . . . . 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 39169 . . . . . . . . . 10 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → ((𝑆‘((𝑘 · 𝐸) + 𝑢))‘((𝑙 · 𝐸) + 𝑣)) = ((𝑙 × (𝐺𝑘)) ((𝑆𝑢)‘𝑣)))
5954, 57, 583eqtr4d 2869 . . . . . . . . 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 1165 . . . . . . . 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 1129 . . . . . . 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 1135 . . . . . . . . . 10 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → 𝑌 = ((𝑙 · 𝐸) + 𝑣))
6362fveq2d 6665 . . . . . . . . 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 1202 . . . . . . . . 9 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → 𝑋 = ((𝑘 · 𝐸) + 𝑢))
6563, 64fveq12d 6668 . . . . . . . 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 6665 . . . . . . 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 6665 . . . . . . . 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 6668 . . . . . . 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 2869 . . . . . 6 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → (𝐺‘((𝑆𝑌)‘𝑋)) = ((𝑆𝑋)‘𝑌))
70693exp 1116 . . . . 5 ((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) → ((𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵) → (𝑌 = ((𝑙 · 𝐸) + 𝑣) → (𝐺‘((𝑆𝑌)‘𝑋)) = ((𝑆𝑋)‘𝑌))))
7170rexlimdvv 3285 . . . 4 ((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) → (∃𝑣 ∈ (𝑂‘{𝐸})∃𝑙𝐵 𝑌 = ((𝑙 · 𝐸) + 𝑣) → (𝐺‘((𝑆𝑌)‘𝑋)) = ((𝑆𝑋)‘𝑌)))
72713exp 1116 . . 3 (𝜑 → ((𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) → (𝑋 = ((𝑘 · 𝐸) + 𝑢) → (∃𝑣 ∈ (𝑂‘{𝐸})∃𝑙𝐵 𝑌 = ((𝑙 · 𝐸) + 𝑣) → (𝐺‘((𝑆𝑌)‘𝑋)) = ((𝑆𝑋)‘𝑌)))))
7372rexlimdvv 3285 . 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 49 1 (𝜑 → (𝐺‘((𝑆𝑌)‘𝑋)) = ((𝑆𝑋)‘𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wrex 3134  wss 3919  {csn 4550  cop 4556   I cid 5446  cres 5544  cfv 6343  (class class class)co 7149  Basecbs 16483  +gcplusg 16565  .rcmulr 16566  Scalarcsca 16568   ·𝑠 cvsca 16569  0gc0g 16713  -gcsg 18105  LSSumclsm 18759  Ringcrg 19297  LModclmod 19634  LSpanclspn 19743  HLchlt 36591  LHypclh 37225  LTrncltrn 37342  DVecHcdvh 38319  ocHcoch 38588  HDMapchdma 39033  HGMapchg 39124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612  ax-riotaBAD 36194
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-ot 4559  df-uni 4825  df-int 4863  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-of 7403  df-om 7575  df-1st 7684  df-2nd 7685  df-tpos 7888  df-undef 7935  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-map 8404  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-2 11697  df-3 11698  df-4 11699  df-5 11700  df-6 11701  df-n0 11895  df-z 11979  df-uz 12241  df-fz 12895  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-mulr 16579  df-sca 16581  df-vsca 16582  df-0g 16715  df-mre 16857  df-mrc 16858  df-acs 16860  df-proset 17538  df-poset 17556  df-plt 17568  df-lub 17584  df-glb 17585  df-join 17586  df-meet 17587  df-p0 17649  df-p1 17650  df-lat 17656  df-clat 17718  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-submnd 17957  df-grp 18106  df-minusg 18107  df-sbg 18108  df-subg 18276  df-cntz 18447  df-oppg 18474  df-lsm 18761  df-cmn 18908  df-abl 18909  df-mgp 19240  df-ur 19252  df-ring 19299  df-oppr 19376  df-dvdsr 19394  df-unit 19395  df-invr 19425  df-dvr 19436  df-drng 19504  df-lmod 19636  df-lss 19704  df-lsp 19744  df-lvec 19875  df-lsatoms 36217  df-lshyp 36218  df-lcv 36260  df-lfl 36299  df-lkr 36327  df-ldual 36365  df-oposet 36417  df-ol 36419  df-oml 36420  df-covers 36507  df-ats 36508  df-atl 36539  df-cvlat 36563  df-hlat 36592  df-llines 36739  df-lplanes 36740  df-lvols 36741  df-lines 36742  df-psubsp 36744  df-pmap 36745  df-padd 37037  df-lhyp 37229  df-laut 37230  df-ldil 37345  df-ltrn 37346  df-trl 37400  df-tgrp 37984  df-tendo 37996  df-edring 37998  df-dveca 38244  df-disoa 38270  df-dvech 38320  df-dib 38380  df-dic 38414  df-dih 38470  df-doch 38589  df-djh 38636  df-lcdual 38828  df-mapd 38866  df-hvmap 38998  df-hdmap1 39034  df-hdmap 39035  df-hgmap 39125
This theorem is referenced by:  hdmapg  39171
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