Theorem xpsvsca 16239

Description: Value of the scalar multiplication function in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)

Hypotheses

Ref	Expression
xpssca.t	⊢ 𝑇 = (𝑅 ×s 𝑆)
xpssca.g	⊢ 𝐺 = (Scalar‘𝑅)
xpssca.1	⊢ (𝜑 → 𝑅 ∈ 𝑉)
xpssca.2	⊢ (𝜑 → 𝑆 ∈ 𝑊)
xpsvsca.x	⊢ 𝑋 = (Base‘𝑅)
xpsvsca.y	⊢ 𝑌 = (Base‘𝑆)
xpsvsca.k	⊢ 𝐾 = (Base‘𝐺)
xpsvsca.m	⊢ · = ( ·𝑠 ‘𝑅)
xpsvsca.n	⊢ × = ( ·𝑠 ‘𝑆)
xpsvsca.p	⊢ ∙ = ( ·𝑠 ‘𝑇)
xpsvsca.3	⊢ (𝜑 → 𝐴 ∈ 𝐾)
xpsvsca.4	⊢ (𝜑 → 𝐵 ∈ 𝑋)
xpsvsca.5	⊢ (𝜑 → 𝐶 ∈ 𝑌)
xpsvsca.6	⊢ (𝜑 → (𝐴 · 𝐵) ∈ 𝑋)
xpsvsca.7	⊢ (𝜑 → (𝐴 × 𝐶) ∈ 𝑌)

Assertion

Ref	Expression
xpsvsca	⊢ (𝜑 → (𝐴 ∙ ⟨𝐵, 𝐶⟩) = ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩)

Proof of Theorem xpsvsca

Dummy variables 𝑘 𝑎 𝑥 𝑦 𝑐 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpsvsca.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐾)
2		df-ov 6653	. . . . 5 ⊢ (𝐵(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))𝐶) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐵, 𝐶⟩)
3		xpsvsca.4	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑋)
4		xpsvsca.5	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑌)
5		eqid 2622	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))
6	5	xpsfval 16227	. . . . . 6 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑌) → (𝐵(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))𝐶) = ◡({𝐵} +𝑐 {𝐶}))
7	3, 4, 6	syl2anc 693	. . . . 5 ⊢ (𝜑 → (𝐵(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))𝐶) = ◡({𝐵} +𝑐 {𝐶}))
8	2, 7	syl5eqr 2670	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐵, 𝐶⟩) = ◡({𝐵} +𝑐 {𝐶}))
9		opelxpi 5148	. . . . . 6 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑌) → ⟨𝐵, 𝐶⟩ ∈ (𝑋 × 𝑌))
10	3, 4, 9	syl2anc 693	. . . . 5 ⊢ (𝜑 → ⟨𝐵, 𝐶⟩ ∈ (𝑋 × 𝑌))
11	5	xpsff1o2 16231	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))
12		f1of 6137	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)⟶ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})))
13	11, 12	ax-mp 5	. . . . . 6 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)⟶ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))
14	13	ffvelrni 6358	. . . . 5 ⊢ (⟨𝐵, 𝐶⟩ ∈ (𝑋 × 𝑌) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐵, 𝐶⟩) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})))
15	10, 14	syl 17	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐵, 𝐶⟩) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})))
16	8, 15	eqeltrrd 2702	. . 3 ⊢ (𝜑 → ◡({𝐵} +𝑐 {𝐶}) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})))
17		xpssca.t	. . . . 5 ⊢ 𝑇 = (𝑅 ×s 𝑆)
18		xpsvsca.x	. . . . 5 ⊢ 𝑋 = (Base‘𝑅)
19		xpsvsca.y	. . . . 5 ⊢ 𝑌 = (Base‘𝑆)
20		xpssca.1	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝑉)
21		xpssca.2	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝑊)
22		xpssca.g	. . . . 5 ⊢ 𝐺 = (Scalar‘𝑅)
23		eqid 2622	. . . . 5 ⊢ (𝐺Xs◡({𝑅} +𝑐 {𝑆})) = (𝐺Xs◡({𝑅} +𝑐 {𝑆}))
24	17, 18, 19, 20, 21, 5, 22, 23	xpsval 16232	. . . 4 ⊢ (𝜑 → 𝑇 = (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) “s (𝐺Xs◡({𝑅} +𝑐 {𝑆}))))
25	17, 18, 19, 20, 21, 5, 22, 23	xpslem 16233	. . . 4 ⊢ (𝜑 → ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) = (Base‘(𝐺Xs◡({𝑅} +𝑐 {𝑆}))))
26		f1ocnv 6149	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))–1-1-onto→(𝑋 × 𝑌))
27	11, 26	mp1i 13	. . . . 5 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))–1-1-onto→(𝑋 × 𝑌))
28		f1ofo 6144	. . . . 5 ⊢ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))–1-1-onto→(𝑋 × 𝑌) → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))–onto→(𝑋 × 𝑌))
29	27, 28	syl 17	. . . 4 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))–onto→(𝑋 × 𝑌))
30		ovexd 6680	. . . 4 ⊢ (𝜑 → (𝐺Xs◡({𝑅} +𝑐 {𝑆})) ∈ V)
31		fvex 6201	. . . . . . . 8 ⊢ (Scalar‘𝑅) ∈ V
32	22, 31	eqeltri 2697	. . . . . . 7 ⊢ 𝐺 ∈ V
33	32	a1i 11	. . . . . 6 ⊢ (⊤ → 𝐺 ∈ V)
34		ovex 6678	. . . . . . . 8 ⊢ ({𝑅} +𝑐 {𝑆}) ∈ V
35	34	cnvex 7113	. . . . . . 7 ⊢ ◡({𝑅} +𝑐 {𝑆}) ∈ V
36	35	a1i 11	. . . . . 6 ⊢ (⊤ → ◡({𝑅} +𝑐 {𝑆}) ∈ V)
37	23, 33, 36	prdssca 16116	. . . . 5 ⊢ (⊤ → 𝐺 = (Scalar‘(𝐺Xs◡({𝑅} +𝑐 {𝑆}))))
38	37	trud 1493	. . . 4 ⊢ 𝐺 = (Scalar‘(𝐺Xs◡({𝑅} +𝑐 {𝑆})))
39		xpsvsca.k	. . . 4 ⊢ 𝐾 = (Base‘𝐺)
40		eqid 2622	. . . 4 ⊢ ( ·𝑠 ‘(𝐺Xs◡({𝑅} +𝑐 {𝑆}))) = ( ·𝑠 ‘(𝐺Xs◡({𝑅} +𝑐 {𝑆})))
41		xpsvsca.p	. . . 4 ⊢ ∙ = ( ·𝑠 ‘𝑇)
42	27	f1ovscpbl 16186	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐾 ∧ 𝑏 ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) ∧ 𝑐 ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})))) → ((◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘𝑏) = (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘𝑐) → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘(𝑎( ·𝑠 ‘(𝐺Xs◡({𝑅} +𝑐 {𝑆})))𝑏)) = (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘(𝑎( ·𝑠 ‘(𝐺Xs◡({𝑅} +𝑐 {𝑆})))𝑐))))
43	24, 25, 29, 30, 38, 39, 40, 41, 42	imasvscaval 16198	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐾 ∧ ◡({𝐵} +𝑐 {𝐶}) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))) → (𝐴 ∙ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({𝐵} +𝑐 {𝐶}))) = (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘(𝐴( ·𝑠 ‘(𝐺Xs◡({𝑅} +𝑐 {𝑆})))◡({𝐵} +𝑐 {𝐶}))))
44	1, 16, 43	mpd3an23 1426	. 2 ⊢ (𝜑 → (𝐴 ∙ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({𝐵} +𝑐 {𝐶}))) = (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘(𝐴( ·𝑠 ‘(𝐺Xs◡({𝑅} +𝑐 {𝑆})))◡({𝐵} +𝑐 {𝐶}))))
45		f1ocnvfv 6534	. . . . 5 ⊢ (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) ∧ ⟨𝐵, 𝐶⟩ ∈ (𝑋 × 𝑌)) → (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐵, 𝐶⟩) = ◡({𝐵} +𝑐 {𝐶}) → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({𝐵} +𝑐 {𝐶})) = ⟨𝐵, 𝐶⟩))
46	11, 10, 45	sylancr 695	. . . 4 ⊢ (𝜑 → (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐵, 𝐶⟩) = ◡({𝐵} +𝑐 {𝐶}) → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({𝐵} +𝑐 {𝐶})) = ⟨𝐵, 𝐶⟩))
47	8, 46	mpd 15	. . 3 ⊢ (𝜑 → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({𝐵} +𝑐 {𝐶})) = ⟨𝐵, 𝐶⟩)
48	47	oveq2d 6666	. 2 ⊢ (𝜑 → (𝐴 ∙ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({𝐵} +𝑐 {𝐶}))) = (𝐴 ∙ ⟨𝐵, 𝐶⟩))
49		iftrue 4092	. . . . . . . . . . . 12 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, 𝑅, 𝑆) = 𝑅)
50	49	fveq2d 6195	. . . . . . . . . . 11 ⊢ (𝑘 = ∅ → ( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆)) = ( ·𝑠 ‘𝑅))
51		xpsvsca.m	. . . . . . . . . . 11 ⊢ · = ( ·𝑠 ‘𝑅)
52	50, 51	syl6eqr 2674	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → ( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆)) = · )
53		eqidd 2623	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → 𝐴 = 𝐴)
54		iftrue 4092	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, 𝐵, 𝐶) = 𝐵)
55	52, 53, 54	oveq123d 6671	. . . . . . . . 9 ⊢ (𝑘 = ∅ → (𝐴( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐵, 𝐶)) = (𝐴 · 𝐵))
56		iftrue 4092	. . . . . . . . 9 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)) = (𝐴 · 𝐵))
57	55, 56	eqtr4d 2659	. . . . . . . 8 ⊢ (𝑘 = ∅ → (𝐴( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐵, 𝐶)) = if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)))
58		iffalse 4095	. . . . . . . . . . . 12 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, 𝑅, 𝑆) = 𝑆)
59	58	fveq2d 6195	. . . . . . . . . . 11 ⊢ (¬ 𝑘 = ∅ → ( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆)) = ( ·𝑠 ‘𝑆))
60		xpsvsca.n	. . . . . . . . . . 11 ⊢ × = ( ·𝑠 ‘𝑆)
61	59, 60	syl6eqr 2674	. . . . . . . . . 10 ⊢ (¬ 𝑘 = ∅ → ( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆)) = × )
62		eqidd 2623	. . . . . . . . . 10 ⊢ (¬ 𝑘 = ∅ → 𝐴 = 𝐴)
63		iffalse 4095	. . . . . . . . . 10 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, 𝐵, 𝐶) = 𝐶)
64	61, 62, 63	oveq123d 6671	. . . . . . . . 9 ⊢ (¬ 𝑘 = ∅ → (𝐴( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐵, 𝐶)) = (𝐴 × 𝐶))
65		iffalse 4095	. . . . . . . . 9 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)) = (𝐴 × 𝐶))
66	64, 65	eqtr4d 2659	. . . . . . . 8 ⊢ (¬ 𝑘 = ∅ → (𝐴( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐵, 𝐶)) = if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)))
67	57, 66	pm2.61i 176	. . . . . . 7 ⊢ (𝐴( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐵, 𝐶)) = if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶))
68	20	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 2𝑜) → 𝑅 ∈ 𝑉)
69	21	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 2𝑜) → 𝑆 ∈ 𝑊)
70		simpr 477	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 2𝑜) → 𝑘 ∈ 2𝑜)
71		xpscfv 16222	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝑘 ∈ 2𝑜) → (◡({𝑅} +𝑐 {𝑆})‘𝑘) = if(𝑘 = ∅, 𝑅, 𝑆))
72	68, 69, 70, 71	syl3anc 1326	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2𝑜) → (◡({𝑅} +𝑐 {𝑆})‘𝑘) = if(𝑘 = ∅, 𝑅, 𝑆))
73	72	fveq2d 6195	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2𝑜) → ( ·𝑠 ‘(◡({𝑅} +𝑐 {𝑆})‘𝑘)) = ( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆)))
74		eqidd 2623	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2𝑜) → 𝐴 = 𝐴)
75	3	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2𝑜) → 𝐵 ∈ 𝑋)
76	4	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2𝑜) → 𝐶 ∈ 𝑌)
77		xpscfv 16222	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑌 ∧ 𝑘 ∈ 2𝑜) → (◡({𝐵} +𝑐 {𝐶})‘𝑘) = if(𝑘 = ∅, 𝐵, 𝐶))
78	75, 76, 70, 77	syl3anc 1326	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2𝑜) → (◡({𝐵} +𝑐 {𝐶})‘𝑘) = if(𝑘 = ∅, 𝐵, 𝐶))
79	73, 74, 78	oveq123d 6671	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 2𝑜) → (𝐴( ·𝑠 ‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐵} +𝑐 {𝐶})‘𝑘)) = (𝐴( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐵, 𝐶)))
80		xpsvsca.6	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ 𝑋)
81	80	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2𝑜) → (𝐴 · 𝐵) ∈ 𝑋)
82		xpsvsca.7	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 × 𝐶) ∈ 𝑌)
83	82	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2𝑜) → (𝐴 × 𝐶) ∈ 𝑌)
84		xpscfv 16222	. . . . . . . 8 ⊢ (((𝐴 · 𝐵) ∈ 𝑋 ∧ (𝐴 × 𝐶) ∈ 𝑌 ∧ 𝑘 ∈ 2𝑜) → (◡({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)})‘𝑘) = if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)))
85	81, 83, 70, 84	syl3anc 1326	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 2𝑜) → (◡({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)})‘𝑘) = if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)))
86	67, 79, 85	3eqtr4a 2682	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 2𝑜) → (𝐴( ·𝑠 ‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐵} +𝑐 {𝐶})‘𝑘)) = (◡({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)})‘𝑘))
87	86	mpteq2dva 4744	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 2𝑜 ↦ (𝐴( ·𝑠 ‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐵} +𝑐 {𝐶})‘𝑘))) = (𝑘 ∈ 2𝑜 ↦ (◡({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)})‘𝑘)))
88		eqid 2622	. . . . . 6 ⊢ (Base‘(𝐺Xs◡({𝑅} +𝑐 {𝑆}))) = (Base‘(𝐺Xs◡({𝑅} +𝑐 {𝑆})))
89	32	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ V)
90		2on 7568	. . . . . . 7 ⊢ 2𝑜 ∈ On
91	90	a1i 11	. . . . . 6 ⊢ (𝜑 → 2𝑜 ∈ On)
92		xpscfn 16219	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ◡({𝑅} +𝑐 {𝑆}) Fn 2𝑜)
93	20, 21, 92	syl2anc 693	. . . . . 6 ⊢ (𝜑 → ◡({𝑅} +𝑐 {𝑆}) Fn 2𝑜)
94	16, 25	eleqtrd 2703	. . . . . 6 ⊢ (𝜑 → ◡({𝐵} +𝑐 {𝐶}) ∈ (Base‘(𝐺Xs◡({𝑅} +𝑐 {𝑆}))))
95	23, 88, 40, 39, 89, 91, 93, 1, 94	prdsvscaval 16139	. . . . 5 ⊢ (𝜑 → (𝐴( ·𝑠 ‘(𝐺Xs◡({𝑅} +𝑐 {𝑆})))◡({𝐵} +𝑐 {𝐶})) = (𝑘 ∈ 2𝑜 ↦ (𝐴( ·𝑠 ‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐵} +𝑐 {𝐶})‘𝑘))))
96		xpscfn 16219	. . . . . . 7 ⊢ (((𝐴 · 𝐵) ∈ 𝑋 ∧ (𝐴 × 𝐶) ∈ 𝑌) → ◡({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}) Fn 2𝑜)
97	80, 82, 96	syl2anc 693	. . . . . 6 ⊢ (𝜑 → ◡({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}) Fn 2𝑜)
98		dffn5 6241	. . . . . 6 ⊢ (◡({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}) Fn 2𝑜 ↔ ◡({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}) = (𝑘 ∈ 2𝑜 ↦ (◡({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)})‘𝑘)))
99	97, 98	sylib 208	. . . . 5 ⊢ (𝜑 → ◡({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}) = (𝑘 ∈ 2𝑜 ↦ (◡({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)})‘𝑘)))
100	87, 95, 99	3eqtr4d 2666	. . . 4 ⊢ (𝜑 → (𝐴( ·𝑠 ‘(𝐺Xs◡({𝑅} +𝑐 {𝑆})))◡({𝐵} +𝑐 {𝐶})) = ◡({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}))
101	100	fveq2d 6195	. . 3 ⊢ (𝜑 → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘(𝐴( ·𝑠 ‘(𝐺Xs◡({𝑅} +𝑐 {𝑆})))◡({𝐵} +𝑐 {𝐶}))) = (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)})))
102		df-ov 6653	. . . . 5 ⊢ ((𝐴 · 𝐵)(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))(𝐴 × 𝐶)) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩)
103	5	xpsfval 16227	. . . . . 6 ⊢ (((𝐴 · 𝐵) ∈ 𝑋 ∧ (𝐴 × 𝐶) ∈ 𝑌) → ((𝐴 · 𝐵)(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))(𝐴 × 𝐶)) = ◡({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}))
104	80, 82, 103	syl2anc 693	. . . . 5 ⊢ (𝜑 → ((𝐴 · 𝐵)(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))(𝐴 × 𝐶)) = ◡({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}))
105	102, 104	syl5eqr 2670	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩) = ◡({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}))
106		opelxpi 5148	. . . . . 6 ⊢ (((𝐴 · 𝐵) ∈ 𝑋 ∧ (𝐴 × 𝐶) ∈ 𝑌) → ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩ ∈ (𝑋 × 𝑌))
107	80, 82, 106	syl2anc 693	. . . . 5 ⊢ (𝜑 → ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩ ∈ (𝑋 × 𝑌))
108		f1ocnvfv 6534	. . . . 5 ⊢ (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) ∧ ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩ ∈ (𝑋 × 𝑌)) → (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩) = ◡({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}) → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)})) = ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩))
109	11, 107, 108	sylancr 695	. . . 4 ⊢ (𝜑 → (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩) = ◡({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}) → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)})) = ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩))
110	105, 109	mpd 15	. . 3 ⊢ (𝜑 → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)})) = ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩)
111	101, 110	eqtrd 2656	. 2 ⊢ (𝜑 → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘(𝐴( ·𝑠 ‘(𝐺Xs◡({𝑅} +𝑐 {𝑆})))◡({𝐵} +𝑐 {𝐶}))) = ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩)
112	44, 48, 111	3eqtr3d 2664	1 ⊢ (𝜑 → (𝐴 ∙ ⟨𝐵, 𝐶⟩) = ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483  ⊤wtru 1484   ∈ wcel 1990  Vcvv 3200  ∅c0 3915  ifcif 4086  {csn 4177  ⟨cop 4183   ↦ cmpt 4729   × cxp 5112  ^◡ccnv 5113  ran crn 5115  Oncon0 5723   Fn wfn 5883  ⟶wf 5884  –onto→wfo 5886  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  2_𝑜c2o 7554   +_𝑐 ccda 8989  Basecbs 15857  Scalarcsca 15944   ·_𝑠 cvsca 15945  X_scprds 16106   ×_s cxps 16166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-prds 16108  df-imas 16168  df-xps 16170

This theorem is referenced by: (None)