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Theorem hdmapglem7 41912
Description: Lemma for hdmapg 41913. 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, and 𝑣 correspond respectively to Baer's w, H, x, y, x', x'', y', and 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 41910 . 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 41910 . 2 (𝜑 → ∃𝑣 ∈ (𝑂‘{𝐸})∃𝑙𝐵 𝑌 = ((𝑙 · 𝐸) + 𝑣))
17 hdmapglem7.c . . . . . . . . . . . 12 = (+g𝑅)
18 hdmapglem7.g . . . . . . . . . . . 12 𝐺 = ((HGMap‘𝐾)‘𝑊)
1912ad2antrr 726 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
201, 4, 12dvhlmod 41093 . . . . . . . . . . . . . . 15 (𝜑𝑈 ∈ LMod)
218lmodring 20883 . . . . . . . . . . . . . . 15 (𝑈 ∈ LMod → 𝑅 ∈ Ring)
2220, 21syl 17 . . . . . . . . . . . . . 14 (𝜑𝑅 ∈ Ring)
2322ad2antrr 726 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → 𝑅 ∈ Ring)
24 simplrr 778 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → 𝑘𝐵)
25 simprr 773 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → 𝑙𝐵)
261, 4, 8, 9, 18, 19, 25hgmapcl 41872 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐺𝑙) ∈ 𝐵)
27 hdmapglem7.t . . . . . . . . . . . . . 14 × = (.r𝑅)
289, 27ringcl 20268 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑘𝐵 ∧ (𝐺𝑙) ∈ 𝐵) → (𝑘 × (𝐺𝑙)) ∈ 𝐵)
2923, 24, 26, 28syl3anc 1370 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝑘 × (𝐺𝑙)) ∈ 𝐵)
30 hdmapglem7.s . . . . . . . . . . . . 13 𝑆 = ((HDMap‘𝐾)‘𝑊)
31 eqid 2735 . . . . . . . . . . . . . . . . . . 19 (Base‘𝐾) = (Base‘𝐾)
32 eqid 2735 . . . . . . . . . . . . . . . . . . 19 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
33 eqid 2735 . . . . . . . . . . . . . . . . . . 19 (0g𝑈) = (0g𝑈)
341, 31, 32, 4, 5, 33, 2, 12dvheveccl 41095 . . . . . . . . . . . . . . . . . 18 (𝜑𝐸 ∈ (𝑉 ∖ {(0g𝑈)}))
3534eldifad 3975 . . . . . . . . . . . . . . . . 17 (𝜑𝐸𝑉)
3635snssd 4814 . . . . . . . . . . . . . . . 16 (𝜑 → {𝐸} ⊆ 𝑉)
371, 4, 5, 3dochssv 41338 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ {𝐸} ⊆ 𝑉) → (𝑂‘{𝐸}) ⊆ 𝑉)
3812, 36, 37syl2anc 584 . . . . . . . . . . . . . . 15 (𝜑 → (𝑂‘{𝐸}) ⊆ 𝑉)
3938ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝑂‘{𝐸}) ⊆ 𝑉)
40 simplrl 777 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → 𝑢 ∈ (𝑂‘{𝐸}))
4139, 40sseldd 3996 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → 𝑢𝑉)
42 simprl 771 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → 𝑣 ∈ (𝑂‘{𝐸}))
4339, 42sseldd 3996 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → 𝑣𝑉)
441, 4, 5, 8, 9, 30, 19, 41, 43hdmapipcl 41888 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → ((𝑆𝑣)‘𝑢) ∈ 𝐵)
451, 4, 8, 9, 17, 18, 19, 29, 44hgmapadd 41877 . . . . . . . . . . 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 41878 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐺‘(𝑘 × (𝐺𝑙))) = ((𝐺‘(𝐺𝑙)) × (𝐺𝑘)))
471, 4, 8, 9, 18, 19, 25hgmapvv 41909 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐺‘(𝐺𝑙)) = 𝑙)
4847oveq1d 7446 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → ((𝐺‘(𝐺𝑙)) × (𝐺𝑘)) = (𝑙 × (𝐺𝑘)))
4946, 48eqtrd 2775 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐺‘(𝑘 × (𝐺𝑙))) = (𝑙 × (𝐺𝑘)))
50 eqid 2735 . . . . . . . . . . . . 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 41905 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐺‘((𝑆𝑣)‘𝑢)) = ((𝑆𝑢)‘𝑣))
5349, 52oveq12d 7449 . . . . . . . . . . 11 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → ((𝐺‘(𝑘 × (𝐺𝑙))) (𝐺‘((𝑆𝑣)‘𝑢))) = ((𝑙 × (𝐺𝑘)) ((𝑆𝑢)‘𝑣)))
5445, 53eqtrd 2775 . . . . . . . . . 10 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐺‘((𝑘 × (𝐺𝑙)) ((𝑆𝑣)‘𝑢))) = ((𝑙 × (𝐺𝑘)) ((𝑆𝑢)‘𝑣)))
5513ad2antrr 726 . . . . . . . . . . . 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 41911 . . . . . . . . . . 11 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → ((𝑆‘((𝑙 · 𝐸) + 𝑣))‘((𝑘 · 𝐸) + 𝑢)) = ((𝑘 × (𝐺𝑙)) ((𝑆𝑣)‘𝑢)))
5756fveq2d 6911 . . . . . . . . . 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 41911 . . . . . . . . . 10 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → ((𝑆‘((𝑘 · 𝐸) + 𝑢))‘((𝑙 · 𝐸) + 𝑣)) = ((𝑙 × (𝐺𝑘)) ((𝑆𝑢)‘𝑣)))
5954, 57, 583eqtr4d 2785 . . . . . . . . 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 1167 . . . . . . . 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 1131 . . . . . . 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 1137 . . . . . . . . . 10 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → 𝑌 = ((𝑙 · 𝐸) + 𝑣))
6362fveq2d 6911 . . . . . . . . 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 1204 . . . . . . . . 9 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → 𝑋 = ((𝑘 · 𝐸) + 𝑢))
6563, 64fveq12d 6914 . . . . . . . 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 6911 . . . . . . 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 6911 . . . . . . . 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 6914 . . . . . . 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 2785 . . . . . 6 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → (𝐺‘((𝑆𝑌)‘𝑋)) = ((𝑆𝑋)‘𝑌))
70693exp 1118 . . . . 5 ((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) → ((𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵) → (𝑌 = ((𝑙 · 𝐸) + 𝑣) → (𝐺‘((𝑆𝑌)‘𝑋)) = ((𝑆𝑋)‘𝑌))))
7170rexlimdvv 3210 . . . 4 ((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) → (∃𝑣 ∈ (𝑂‘{𝐸})∃𝑙𝐵 𝑌 = ((𝑙 · 𝐸) + 𝑣) → (𝐺‘((𝑆𝑌)‘𝑋)) = ((𝑆𝑋)‘𝑌)))
72713exp 1118 . . 3 (𝜑 → ((𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) → (𝑋 = ((𝑘 · 𝐸) + 𝑢) → (∃𝑣 ∈ (𝑂‘{𝐸})∃𝑙𝐵 𝑌 = ((𝑙 · 𝐸) + 𝑣) → (𝐺‘((𝑆𝑌)‘𝑋)) = ((𝑆𝑋)‘𝑌)))))
7372rexlimdvv 3210 . 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 395  w3a 1086   = wceq 1537  wcel 2106  wrex 3068  wss 3963  {csn 4631  cop 4637   I cid 5582  cres 5691  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  .rcmulr 17299  Scalarcsca 17301   ·𝑠 cvsca 17302  0gc0g 17486  -gcsg 18966  LSSumclsm 19667  Ringcrg 20251  LModclmod 20875  LSpanclspn 20987  HLchlt 39332  LHypclh 39967  LTrncltrn 40084  DVecHcdvh 41061  ocHcoch 41330  HDMapchdma 41775  HGMapchg 41866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-riotaBAD 38935
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-tpos 8250  df-undef 8297  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-0g 17488  df-mre 17631  df-mrc 17632  df-acs 17634  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-p1 18484  df-lat 18490  df-clat 18557  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154  df-cntz 19348  df-oppg 19377  df-lsm 19669  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-oppr 20351  df-dvdsr 20374  df-unit 20375  df-invr 20405  df-dvr 20418  df-nzr 20530  df-rlreg 20711  df-domn 20712  df-drng 20748  df-lmod 20877  df-lss 20948  df-lsp 20988  df-lvec 21120  df-lsatoms 38958  df-lshyp 38959  df-lcv 39001  df-lfl 39040  df-lkr 39068  df-ldual 39106  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-llines 39481  df-lplanes 39482  df-lvols 39483  df-lines 39484  df-psubsp 39486  df-pmap 39487  df-padd 39779  df-lhyp 39971  df-laut 39972  df-ldil 40087  df-ltrn 40088  df-trl 40142  df-tgrp 40726  df-tendo 40738  df-edring 40740  df-dveca 40986  df-disoa 41012  df-dvech 41062  df-dib 41122  df-dic 41156  df-dih 41212  df-doch 41331  df-djh 41378  df-lcdual 41570  df-mapd 41608  df-hvmap 41740  df-hdmap1 41776  df-hdmap 41777  df-hgmap 41867
This theorem is referenced by:  hdmapg  41913
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