hdmapglem7 - Mathbox for Norm Megill

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Theorem hdmapglem7 42189

Description: Lemma for hdmapg 42190. 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 ((𝑆‘𝑌)‘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	⊢ + = (+_g‘𝑈)
hdmapglem7.q	⊢ · = ( ·_𝑠 ‘𝑈)
hdmapglem7.r	⊢ 𝑅 = (Scalar‘𝑈)
hdmapglem7.b	⊢ 𝐵 = (Base‘𝑅)
hdmapglem7.a	⊢ ⊕ = (LSSum‘𝑈)
hdmapglem7.n	⊢ 𝑁 = (LSpan‘𝑈)
hdmapglem7.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmapglem7.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
hdmapglem7.t	⊢ × = (._r‘𝑅)
hdmapglem7.z	⊢ 0 = (0_g‘𝑅)
hdmapglem7.c	⊢ ✚ = (+_g‘𝑅)
hdmapglem7.s	⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmapglem7.g	⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)
hdmapglem7.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)

**Assertion**
Ref	Expression
hdmapglem7	⊢ (𝜑 → (𝐺‘((𝑆‘𝑌)‘𝑋)) = ((𝑆‘𝑋)‘𝑌))

Proof of Theorem hdmapglem7 Dummy variables 𝑘 𝑙 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	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 ⊢ + = (+_g‘𝑈)
7		hdmapglem7.q	. . 3 ⊢ · = ( ·_𝑠 ‘𝑈)
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 ∧ 𝑊 ∈ 𝐻))
13		hdmapglem7.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
14	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	hdmapglem7a 42187	. 2 ⊢ (𝜑 → ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢))
15		hdmapglem7.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15	hdmapglem7a 42187	. 2 ⊢ (𝜑 → ∃𝑣 ∈ (𝑂‘{𝐸})∃𝑙 ∈ 𝐵 𝑌 = ((𝑙 · 𝐸) + 𝑣))
17		hdmapglem7.c	. . . . . . . . . . . 12 ⊢ ✚ = (+_g‘𝑅)
18		hdmapglem7.g	. . . . . . . . . . . 12 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)
19	12	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
20	1, 4, 12	dvhlmod 41370	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑈 ∈ LMod)
21	8	lmodring 20819	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ∈ LMod → 𝑅 ∈ Ring)
22	20, 21	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ Ring)
23	22	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → 𝑅 ∈ Ring)
24		simplrr 777	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → 𝑘 ∈ 𝐵)
25		simprr 772	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → 𝑙 ∈ 𝐵)
26	1, 4, 8, 9, 18, 19, 25	hgmapcl 42149	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → (𝐺‘𝑙) ∈ 𝐵)
27		hdmapglem7.t	. . . . . . . . . . . . . 14 ⊢ × = (._r‘𝑅)
28	9, 27	ringcl 20185	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑘 ∈ 𝐵 ∧ (𝐺‘𝑙) ∈ 𝐵) → (𝑘 × (𝐺‘𝑙)) ∈ 𝐵)
29	23, 24, 26, 28	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → (𝑘 × (𝐺‘𝑙)) ∈ 𝐵)
30		hdmapglem7.s	. . . . . . . . . . . . 13 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
31		eqid 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝐾) = (Base‘𝐾)
32		eqid 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
33		eqid 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
34	1, 31, 32, 4, 5, 33, 2, 12	dvheveccl 41372	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
35	34	eldifad 3913	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐸 ∈ 𝑉)
36	35	snssd 4765	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝐸} ⊆ 𝑉)
37	1, 4, 5, 3	dochssv 41615	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝐸} ⊆ 𝑉) → (𝑂‘{𝐸}) ⊆ 𝑉)
38	12, 36, 37	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑂‘{𝐸}) ⊆ 𝑉)
39	38	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → (𝑂‘{𝐸}) ⊆ 𝑉)
40		simplrl 776	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → 𝑢 ∈ (𝑂‘{𝐸}))
41	39, 40	sseldd 3934	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → 𝑢 ∈ 𝑉)
42		simprl 770	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → 𝑣 ∈ (𝑂‘{𝐸}))
43	39, 42	sseldd 3934	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → 𝑣 ∈ 𝑉)
44	1, 4, 5, 8, 9, 30, 19, 41, 43	hdmapipcl 42165	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → ((𝑆‘𝑣)‘𝑢) ∈ 𝐵)
45	1, 4, 8, 9, 17, 18, 19, 29, 44	hgmapadd 42154	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → (𝐺‘((𝑘 × (𝐺‘𝑙)) ✚ ((𝑆‘𝑣)‘𝑢))) = ((𝐺‘(𝑘 × (𝐺‘𝑙))) ✚ (𝐺‘((𝑆‘𝑣)‘𝑢))))
46	1, 4, 8, 9, 27, 18, 19, 24, 26	hgmapmul 42155	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → (𝐺‘(𝑘 × (𝐺‘𝑙))) = ((𝐺‘(𝐺‘𝑙)) × (𝐺‘𝑘)))
47	1, 4, 8, 9, 18, 19, 25	hgmapvv 42186	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → (𝐺‘(𝐺‘𝑙)) = 𝑙)
48	47	oveq1d 7373	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → ((𝐺‘(𝐺‘𝑙)) × (𝐺‘𝑘)) = (𝑙 × (𝐺‘𝑘)))
49	46, 48	eqtrd 2771	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → (𝐺‘(𝑘 × (𝐺‘𝑙))) = (𝑙 × (𝐺‘𝑘)))
50		eqid 2736	. . . . . . . . . . . . 13 ⊢ (-_g‘𝑈) = (-_g‘𝑈)
51		hdmapglem7.z	. . . . . . . . . . . . 13 ⊢ 0 = (0_g‘𝑅)
52	1, 2, 3, 4, 5, 6, 50, 7, 8, 9, 27, 51, 30, 18, 19, 40, 42, 24, 24	hdmapglem5 42182	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → (𝐺‘((𝑆‘𝑣)‘𝑢)) = ((𝑆‘𝑢)‘𝑣))
53	49, 52	oveq12d 7376	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → ((𝐺‘(𝑘 × (𝐺‘𝑙))) ✚ (𝐺‘((𝑆‘𝑣)‘𝑢))) = ((𝑙 × (𝐺‘𝑘)) ✚ ((𝑆‘𝑢)‘𝑣)))
54	45, 53	eqtrd 2771	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → (𝐺‘((𝑘 × (𝐺‘𝑙)) ✚ ((𝑆‘𝑣)‘𝑢))) = ((𝑙 × (𝐺‘𝑘)) ✚ ((𝑆‘𝑢)‘𝑣)))
55	13	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → 𝑋 ∈ 𝑉)
56	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 19, 55, 27, 51, 17, 30, 18, 42, 40, 25, 24	hdmapglem7b 42188	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → ((𝑆‘((𝑙 · 𝐸) + 𝑣))‘((𝑘 · 𝐸) + 𝑢)) = ((𝑘 × (𝐺‘𝑙)) ✚ ((𝑆‘𝑣)‘𝑢)))
57	56	fveq2d 6838	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → (𝐺‘((𝑆‘((𝑙 · 𝐸) + 𝑣))‘((𝑘 · 𝐸) + 𝑢))) = (𝐺‘((𝑘 × (𝐺‘𝑙)) ✚ ((𝑆‘𝑣)‘𝑢))))
58	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 19, 55, 27, 51, 17, 30, 18, 40, 42, 24, 25	hdmapglem7b 42188	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → ((𝑆‘((𝑘 · 𝐸) + 𝑢))‘((𝑙 · 𝐸) + 𝑣)) = ((𝑙 × (𝐺‘𝑘)) ✚ ((𝑆‘𝑢)‘𝑣)))
59	54, 57, 58	3eqtr4d 2781	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → (𝐺‘((𝑆‘((𝑙 · 𝐸) + 𝑣))‘((𝑘 · 𝐸) + 𝑢))) = ((𝑆‘((𝑘 · 𝐸) + 𝑢))‘((𝑙 · 𝐸) + 𝑣)))
60	59	3adantl3 1169	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → (𝐺‘((𝑆‘((𝑙 · 𝐸) + 𝑣))‘((𝑘 · 𝐸) + 𝑢))) = ((𝑆‘((𝑘 · 𝐸) + 𝑢))‘((𝑙 · 𝐸) + 𝑣)))
61	60	3adant3 1132	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → (𝐺‘((𝑆‘((𝑙 · 𝐸) + 𝑣))‘((𝑘 · 𝐸) + 𝑢))) = ((𝑆‘((𝑘 · 𝐸) + 𝑢))‘((𝑙 · 𝐸) + 𝑣)))
62		simp3 1138	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → 𝑌 = ((𝑙 · 𝐸) + 𝑣))
63	62	fveq2d 6838	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → (𝑆‘𝑌) = (𝑆‘((𝑙 · 𝐸) + 𝑣)))
64		simp13 1206	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → 𝑋 = ((𝑘 · 𝐸) + 𝑢))
65	63, 64	fveq12d 6841	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → ((𝑆‘𝑌)‘𝑋) = ((𝑆‘((𝑙 · 𝐸) + 𝑣))‘((𝑘 · 𝐸) + 𝑢)))
66	65	fveq2d 6838	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → (𝐺‘((𝑆‘𝑌)‘𝑋)) = (𝐺‘((𝑆‘((𝑙 · 𝐸) + 𝑣))‘((𝑘 · 𝐸) + 𝑢))))
67	64	fveq2d 6838	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → (𝑆‘𝑋) = (𝑆‘((𝑘 · 𝐸) + 𝑢)))
68	67, 62	fveq12d 6841	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → ((𝑆‘𝑋)‘𝑌) = ((𝑆‘((𝑘 · 𝐸) + 𝑢))‘((𝑙 · 𝐸) + 𝑣)))
69	61, 66, 68	3eqtr4d 2781	. . . . . 6 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → (𝐺‘((𝑆‘𝑌)‘𝑋)) = ((𝑆‘𝑋)‘𝑌))
70	69	3exp 1119	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) → ((𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵) → (𝑌 = ((𝑙 · 𝐸) + 𝑣) → (𝐺‘((𝑆‘𝑌)‘𝑋)) = ((𝑆‘𝑋)‘𝑌))))
71	70	rexlimdvv 3192	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) → (∃𝑣 ∈ (𝑂‘{𝐸})∃𝑙 ∈ 𝐵 𝑌 = ((𝑙 · 𝐸) + 𝑣) → (𝐺‘((𝑆‘𝑌)‘𝑋)) = ((𝑆‘𝑋)‘𝑌)))
72	71	3exp 1119	. . 3 ⊢ (𝜑 → ((𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵) → (𝑋 = ((𝑘 · 𝐸) + 𝑢) → (∃𝑣 ∈ (𝑂‘{𝐸})∃𝑙 ∈ 𝐵 𝑌 = ((𝑙 · 𝐸) + 𝑣) → (𝐺‘((𝑆‘𝑌)‘𝑋)) = ((𝑆‘𝑋)‘𝑌)))))
73	72	rexlimdvv 3192	. 2 ⊢ (𝜑 → (∃𝑢 ∈ (𝑂‘{𝐸})∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢) → (∃𝑣 ∈ (𝑂‘{𝐸})∃𝑙 ∈ 𝐵 𝑌 = ((𝑙 · 𝐸) + 𝑣) → (𝐺‘((𝑆‘𝑌)‘𝑋)) = ((𝑆‘𝑋)‘𝑌))))
74	14, 16, 73	mp2d 49	1 ⊢ (𝜑 → (𝐺‘((𝑆‘𝑌)‘𝑋)) = ((𝑆‘𝑋)‘𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ∃wrex 3060 ⊆ wss 3901 {csn 4580 ⟨cop 4586 I cid 5518 ↾ cres 5626 ‘cfv 6492 (class class class)co 7358 Basecbs 17136 +_gcplusg 17177 ._rcmulr 17178 Scalarcsca 17180 ·_𝑠 cvsca 17181 0_gc0g 17359 -_gcsg 18865 LSSumclsm 19563 Ringcrg 20168 LModclmod 20811 LSpanclspn 20922 HLchlt 39610 LHypclh 40244 LTrncltrn 40361 DVecHcdvh 41338 ocHcoch 41607 HDMapchdma 42052 HGMapchg 42143

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-riotaBAD 39213

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-ot 4589 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-tpos 8168 df-undef 8215 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-n0 12402 df-z 12489 df-uz 12752 df-fz 13424 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-sca 17193 df-vsca 17194 df-0g 17361 df-mre 17505 df-mrc 17506 df-acs 17508 df-proset 18217 df-poset 18236 df-plt 18251 df-lub 18267 df-glb 18268 df-join 18269 df-meet 18270 df-p0 18346 df-p1 18347 df-lat 18355 df-clat 18422 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19053 df-cntz 19246 df-oppg 19275 df-lsm 19565 df-cmn 19711 df-abl 19712 df-mgp 20076 df-rng 20088 df-ur 20117 df-ring 20170 df-oppr 20273 df-dvdsr 20293 df-unit 20294 df-invr 20324 df-dvr 20337 df-nzr 20446 df-rlreg 20627 df-domn 20628 df-drng 20664 df-lmod 20813 df-lss 20883 df-lsp 20923 df-lvec 21055 df-lsatoms 39236 df-lshyp 39237 df-lcv 39279 df-lfl 39318 df-lkr 39346 df-ldual 39384 df-oposet 39436 df-ol 39438 df-oml 39439 df-covers 39526 df-ats 39527 df-atl 39558 df-cvlat 39582 df-hlat 39611 df-llines 39758 df-lplanes 39759 df-lvols 39760 df-lines 39761 df-psubsp 39763 df-pmap 39764 df-padd 40056 df-lhyp 40248 df-laut 40249 df-ldil 40364 df-ltrn 40365 df-trl 40419 df-tgrp 41003 df-tendo 41015 df-edring 41017 df-dveca 41263 df-disoa 41289 df-dvech 41339 df-dib 41399 df-dic 41433 df-dih 41489 df-doch 41608 df-djh 41655 df-lcdual 41847 df-mapd 41885 df-hvmap 42017 df-hdmap1 42053 df-hdmap 42054 df-hgmap 42144

This theorem is referenced by: hdmapg 42190

W3C validator