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Theorem hdmapglem7 37040
Description: Lemma for hdmapg 37041. 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 37038 . 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 37038 . 2 (𝜑 → ∃𝑣 ∈ (𝑂‘{𝐸})∃𝑙𝐵 𝑌 = ((𝑙 · 𝐸) + 𝑣))
17 hdmapglem7.c . . . . . . . . . . . 12 = (+g𝑅)
18 hdmapglem7.g . . . . . . . . . . . 12 𝐺 = ((HGMap‘𝐾)‘𝑊)
1912ad2antrr 761 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
201, 4, 12dvhlmod 36218 . . . . . . . . . . . . . . 15 (𝜑𝑈 ∈ LMod)
218lmodring 18852 . . . . . . . . . . . . . . 15 (𝑈 ∈ LMod → 𝑅 ∈ Ring)
2220, 21syl 17 . . . . . . . . . . . . . 14 (𝜑𝑅 ∈ Ring)
2322ad2antrr 761 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → 𝑅 ∈ Ring)
24 simplrr 800 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → 𝑘𝐵)
25 simprr 795 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → 𝑙𝐵)
261, 4, 8, 9, 18, 19, 25hgmapcl 37000 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐺𝑙) ∈ 𝐵)
27 hdmapglem7.t . . . . . . . . . . . . . 14 × = (.r𝑅)
289, 27ringcl 18542 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑘𝐵 ∧ (𝐺𝑙) ∈ 𝐵) → (𝑘 × (𝐺𝑙)) ∈ 𝐵)
2923, 24, 26, 28syl3anc 1324 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝑘 × (𝐺𝑙)) ∈ 𝐵)
30 hdmapglem7.s . . . . . . . . . . . . 13 𝑆 = ((HDMap‘𝐾)‘𝑊)
31 eqid 2620 . . . . . . . . . . . . . . . . . . 19 (Base‘𝐾) = (Base‘𝐾)
32 eqid 2620 . . . . . . . . . . . . . . . . . . 19 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
33 eqid 2620 . . . . . . . . . . . . . . . . . . 19 (0g𝑈) = (0g𝑈)
341, 31, 32, 4, 5, 33, 2, 12dvheveccl 36220 . . . . . . . . . . . . . . . . . 18 (𝜑𝐸 ∈ (𝑉 ∖ {(0g𝑈)}))
3534eldifad 3579 . . . . . . . . . . . . . . . . 17 (𝜑𝐸𝑉)
3635snssd 4331 . . . . . . . . . . . . . . . 16 (𝜑 → {𝐸} ⊆ 𝑉)
371, 4, 5, 3dochssv 36463 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ {𝐸} ⊆ 𝑉) → (𝑂‘{𝐸}) ⊆ 𝑉)
3812, 36, 37syl2anc 692 . . . . . . . . . . . . . . 15 (𝜑 → (𝑂‘{𝐸}) ⊆ 𝑉)
3938ad2antrr 761 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝑂‘{𝐸}) ⊆ 𝑉)
40 simplrl 799 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → 𝑢 ∈ (𝑂‘{𝐸}))
4139, 40sseldd 3596 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → 𝑢𝑉)
42 simprl 793 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → 𝑣 ∈ (𝑂‘{𝐸}))
4339, 42sseldd 3596 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → 𝑣𝑉)
441, 4, 5, 8, 9, 30, 19, 41, 43hdmapipcl 37016 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → ((𝑆𝑣)‘𝑢) ∈ 𝐵)
451, 4, 8, 9, 17, 18, 19, 29, 44hgmapadd 37005 . . . . . . . . . . 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 37006 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐺‘(𝑘 × (𝐺𝑙))) = ((𝐺‘(𝐺𝑙)) × (𝐺𝑘)))
471, 4, 8, 9, 18, 19, 25hgmapvv 37037 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐺‘(𝐺𝑙)) = 𝑙)
4847oveq1d 6650 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → ((𝐺‘(𝐺𝑙)) × (𝐺𝑘)) = (𝑙 × (𝐺𝑘)))
4946, 48eqtrd 2654 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐺‘(𝑘 × (𝐺𝑙))) = (𝑙 × (𝐺𝑘)))
50 eqid 2620 . . . . . . . . . . . . 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 37033 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐺‘((𝑆𝑣)‘𝑢)) = ((𝑆𝑢)‘𝑣))
5349, 52oveq12d 6653 . . . . . . . . . . 11 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → ((𝐺‘(𝑘 × (𝐺𝑙))) (𝐺‘((𝑆𝑣)‘𝑢))) = ((𝑙 × (𝐺𝑘)) ((𝑆𝑢)‘𝑣)))
5445, 53eqtrd 2654 . . . . . . . . . 10 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐺‘((𝑘 × (𝐺𝑙)) ((𝑆𝑣)‘𝑢))) = ((𝑙 × (𝐺𝑘)) ((𝑆𝑢)‘𝑣)))
5513ad2antrr 761 . . . . . . . . . . . 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 37039 . . . . . . . . . . 11 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → ((𝑆‘((𝑙 · 𝐸) + 𝑣))‘((𝑘 · 𝐸) + 𝑢)) = ((𝑘 × (𝐺𝑙)) ((𝑆𝑣)‘𝑢)))
5756fveq2d 6182 . . . . . . . . . 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 37039 . . . . . . . . . 10 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → ((𝑆‘((𝑘 · 𝐸) + 𝑢))‘((𝑙 · 𝐸) + 𝑣)) = ((𝑙 × (𝐺𝑘)) ((𝑆𝑢)‘𝑣)))
5954, 57, 583eqtr4d 2664 . . . . . . . . 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 1217 . . . . . . . 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 1079 . . . . . . 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 1061 . . . . . . . . . 10 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → 𝑌 = ((𝑙 · 𝐸) + 𝑣))
6362fveq2d 6182 . . . . . . . . 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 1091 . . . . . . . . 9 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → 𝑋 = ((𝑘 · 𝐸) + 𝑢))
6563, 64fveq12d 6184 . . . . . . . 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 6182 . . . . . . 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 6182 . . . . . . . 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 6184 . . . . . . 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 2664 . . . . . 6 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → (𝐺‘((𝑆𝑌)‘𝑋)) = ((𝑆𝑋)‘𝑌))
70693exp 1262 . . . . 5 ((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) → ((𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵) → (𝑌 = ((𝑙 · 𝐸) + 𝑣) → (𝐺‘((𝑆𝑌)‘𝑋)) = ((𝑆𝑋)‘𝑌))))
7170rexlimdvv 3033 . . . 4 ((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) → (∃𝑣 ∈ (𝑂‘{𝐸})∃𝑙𝐵 𝑌 = ((𝑙 · 𝐸) + 𝑣) → (𝐺‘((𝑆𝑌)‘𝑋)) = ((𝑆𝑋)‘𝑌)))
72713exp 1262 . . 3 (𝜑 → ((𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) → (𝑋 = ((𝑘 · 𝐸) + 𝑢) → (∃𝑣 ∈ (𝑂‘{𝐸})∃𝑙𝐵 𝑌 = ((𝑙 · 𝐸) + 𝑣) → (𝐺‘((𝑆𝑌)‘𝑋)) = ((𝑆𝑋)‘𝑌)))))
7372rexlimdvv 3033 . 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 384  w3a 1036   = wceq 1481  wcel 1988  wrex 2910  wss 3567  {csn 4168  cop 4174   I cid 5013  cres 5106  cfv 5876  (class class class)co 6635  Basecbs 15838  +gcplusg 15922  .rcmulr 15923  Scalarcsca 15925   ·𝑠 cvsca 15926  0gc0g 16081  -gcsg 17405  LSSumclsm 18030  Ringcrg 18528  LModclmod 18844  LSpanclspn 18952  HLchlt 34456  LHypclh 35089  LTrncltrn 35206  DVecHcdvh 36186  ocHcoch 36455  HDMapchdma 36901  HGMapchg 36994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998  ax-riotaBAD 34058
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-ot 4177  df-uni 4428  df-int 4467  df-iun 4513  df-iin 4514  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-of 6882  df-om 7051  df-1st 7153  df-2nd 7154  df-tpos 7337  df-undef 7384  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-3 11065  df-4 11066  df-5 11067  df-6 11068  df-n0 11278  df-z 11363  df-uz 11673  df-fz 12312  df-struct 15840  df-ndx 15841  df-slot 15842  df-base 15844  df-sets 15845  df-ress 15846  df-plusg 15935  df-mulr 15936  df-sca 15938  df-vsca 15939  df-0g 16083  df-mre 16227  df-mrc 16228  df-acs 16230  df-preset 16909  df-poset 16927  df-plt 16939  df-lub 16955  df-glb 16956  df-join 16957  df-meet 16958  df-p0 17020  df-p1 17021  df-lat 17027  df-clat 17089  df-mgm 17223  df-sgrp 17265  df-mnd 17276  df-submnd 17317  df-grp 17406  df-minusg 17407  df-sbg 17408  df-subg 17572  df-cntz 17731  df-oppg 17757  df-lsm 18032  df-cmn 18176  df-abl 18177  df-mgp 18471  df-ur 18483  df-ring 18530  df-oppr 18604  df-dvdsr 18622  df-unit 18623  df-invr 18653  df-dvr 18664  df-drng 18730  df-lmod 18846  df-lss 18914  df-lsp 18953  df-lvec 19084  df-lsatoms 34082  df-lshyp 34083  df-lcv 34125  df-lfl 34164  df-lkr 34192  df-ldual 34230  df-oposet 34282  df-ol 34284  df-oml 34285  df-covers 34372  df-ats 34373  df-atl 34404  df-cvlat 34428  df-hlat 34457  df-llines 34603  df-lplanes 34604  df-lvols 34605  df-lines 34606  df-psubsp 34608  df-pmap 34609  df-padd 34901  df-lhyp 35093  df-laut 35094  df-ldil 35209  df-ltrn 35210  df-trl 35265  df-tgrp 35850  df-tendo 35862  df-edring 35864  df-dveca 36110  df-disoa 36137  df-dvech 36187  df-dib 36247  df-dic 36281  df-dih 36337  df-doch 36456  df-djh 36503  df-lcdual 36695  df-mapd 36733  df-hvmap 36865  df-hdmap1 36902  df-hdmap 36903  df-hgmap 36995
This theorem is referenced by:  hdmapg  37041
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