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Theorem hdmapglem7 41931
Description: Lemma for hdmapg 41932. 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 41929 . 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 41929 . 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 41112 . . . . . . . . . . . . . . 15 (𝜑𝑈 ∈ LMod)
218lmodring 20866 . . . . . . . . . . . . . . 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 41891 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐺𝑙) ∈ 𝐵)
27 hdmapglem7.t . . . . . . . . . . . . . 14 × = (.r𝑅)
289, 27ringcl 20247 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑘𝐵 ∧ (𝐺𝑙) ∈ 𝐵) → (𝑘 × (𝐺𝑙)) ∈ 𝐵)
2923, 24, 26, 28syl3anc 1373 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝑘 × (𝐺𝑙)) ∈ 𝐵)
30 hdmapglem7.s . . . . . . . . . . . . 13 𝑆 = ((HDMap‘𝐾)‘𝑊)
31 eqid 2737 . . . . . . . . . . . . . . . . . . 19 (Base‘𝐾) = (Base‘𝐾)
32 eqid 2737 . . . . . . . . . . . . . . . . . . 19 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
33 eqid 2737 . . . . . . . . . . . . . . . . . . 19 (0g𝑈) = (0g𝑈)
341, 31, 32, 4, 5, 33, 2, 12dvheveccl 41114 . . . . . . . . . . . . . . . . . 18 (𝜑𝐸 ∈ (𝑉 ∖ {(0g𝑈)}))
3534eldifad 3963 . . . . . . . . . . . . . . . . 17 (𝜑𝐸𝑉)
3635snssd 4809 . . . . . . . . . . . . . . . 16 (𝜑 → {𝐸} ⊆ 𝑉)
371, 4, 5, 3dochssv 41357 . . . . . . . . . . . . . . . 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 3984 . . . . . . . . . . . . 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 3984 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → 𝑣𝑉)
441, 4, 5, 8, 9, 30, 19, 41, 43hdmapipcl 41907 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → ((𝑆𝑣)‘𝑢) ∈ 𝐵)
451, 4, 8, 9, 17, 18, 19, 29, 44hgmapadd 41896 . . . . . . . . . . 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 41897 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐺‘(𝑘 × (𝐺𝑙))) = ((𝐺‘(𝐺𝑙)) × (𝐺𝑘)))
471, 4, 8, 9, 18, 19, 25hgmapvv 41928 . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → (𝐺‘(𝑘 × (𝐺𝑙))) = (𝑙 × (𝐺𝑘)))
50 eqid 2737 . . . . . . . . . . . . 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 41924 . . . . . . . . . . . 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 2777 . . . . . . . . . 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 41930 . . . . . . . . . . 11 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → ((𝑆‘((𝑙 · 𝐸) + 𝑣))‘((𝑘 · 𝐸) + 𝑢)) = ((𝑘 × (𝐺𝑙)) ((𝑆𝑣)‘𝑢)))
5756fveq2d 6910 . . . . . . . . . 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 41930 . . . . . . . . . 10 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵)) → ((𝑆‘((𝑘 · 𝐸) + 𝑢))‘((𝑙 · 𝐸) + 𝑣)) = ((𝑙 × (𝐺𝑘)) ((𝑆𝑢)‘𝑣)))
5954, 57, 583eqtr4d 2787 . . . . . . . . 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 1169 . . . . . . . 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 1133 . . . . . . 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 1139 . . . . . . . . . 10 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → 𝑌 = ((𝑙 · 𝐸) + 𝑣))
6362fveq2d 6910 . . . . . . . . 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 1206 . . . . . . . . 9 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → 𝑋 = ((𝑘 · 𝐸) + 𝑢))
6563, 64fveq12d 6913 . . . . . . . 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 6910 . . . . . . 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 6910 . . . . . . . 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 6913 . . . . . . 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 2787 . . . . . 6 (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → (𝐺‘((𝑆𝑌)‘𝑋)) = ((𝑆𝑋)‘𝑌))
70693exp 1120 . . . . 5 ((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) → ((𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙𝐵) → (𝑌 = ((𝑙 · 𝐸) + 𝑣) → (𝐺‘((𝑆𝑌)‘𝑋)) = ((𝑆𝑋)‘𝑌))))
7170rexlimdvv 3212 . . . 4 ((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) → (∃𝑣 ∈ (𝑂‘{𝐸})∃𝑙𝐵 𝑌 = ((𝑙 · 𝐸) + 𝑣) → (𝐺‘((𝑆𝑌)‘𝑋)) = ((𝑆𝑋)‘𝑌)))
72713exp 1120 . . 3 (𝜑 → ((𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘𝐵) → (𝑋 = ((𝑘 · 𝐸) + 𝑢) → (∃𝑣 ∈ (𝑂‘{𝐸})∃𝑙𝐵 𝑌 = ((𝑙 · 𝐸) + 𝑣) → (𝐺‘((𝑆𝑌)‘𝑋)) = ((𝑆𝑋)‘𝑌)))))
7372rexlimdvv 3212 . 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 1087   = wceq 1540  wcel 2108  wrex 3070  wss 3951  {csn 4626  cop 4632   I cid 5577  cres 5687  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  .rcmulr 17298  Scalarcsca 17300   ·𝑠 cvsca 17301  0gc0g 17484  -gcsg 18953  LSSumclsm 19652  Ringcrg 20230  LModclmod 20858  LSpanclspn 20969  HLchlt 39351  LHypclh 39986  LTrncltrn 40103  DVecHcdvh 41080  ocHcoch 41349  HDMapchdma 41794  HGMapchg 41885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-riotaBAD 38954
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-tpos 8251  df-undef 8298  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-0g 17486  df-mre 17629  df-mrc 17630  df-acs 17632  df-proset 18340  df-poset 18359  df-plt 18375  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-p1 18471  df-lat 18477  df-clat 18544  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-subg 19141  df-cntz 19335  df-oppg 19364  df-lsm 19654  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-oppr 20334  df-dvdsr 20357  df-unit 20358  df-invr 20388  df-dvr 20401  df-nzr 20513  df-rlreg 20694  df-domn 20695  df-drng 20731  df-lmod 20860  df-lss 20930  df-lsp 20970  df-lvec 21102  df-lsatoms 38977  df-lshyp 38978  df-lcv 39020  df-lfl 39059  df-lkr 39087  df-ldual 39125  df-oposet 39177  df-ol 39179  df-oml 39180  df-covers 39267  df-ats 39268  df-atl 39299  df-cvlat 39323  df-hlat 39352  df-llines 39500  df-lplanes 39501  df-lvols 39502  df-lines 39503  df-psubsp 39505  df-pmap 39506  df-padd 39798  df-lhyp 39990  df-laut 39991  df-ldil 40106  df-ltrn 40107  df-trl 40161  df-tgrp 40745  df-tendo 40757  df-edring 40759  df-dveca 41005  df-disoa 41031  df-dvech 41081  df-dib 41141  df-dic 41175  df-dih 41231  df-doch 41350  df-djh 41397  df-lcdual 41589  df-mapd 41627  df-hvmap 41759  df-hdmap1 41795  df-hdmap 41796  df-hgmap 41886
This theorem is referenced by:  hdmapg  41932
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