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Theorem xpsvsca 16441
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 (𝜑𝑅𝑉)
xpssca.2 (𝜑𝑆𝑊)
xpsvsca.x 𝑋 = (Base‘𝑅)
xpsvsca.y 𝑌 = (Base‘𝑆)
xpsvsca.k 𝐾 = (Base‘𝐺)
xpsvsca.m · = ( ·𝑠𝑅)
xpsvsca.n × = ( ·𝑠𝑆)
xpsvsca.p = ( ·𝑠𝑇)
xpsvsca.3 (𝜑𝐴𝐾)
xpsvsca.4 (𝜑𝐵𝑋)
xpsvsca.5 (𝜑𝐶𝑌)
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.
StepHypRef Expression
1 xpsvsca.3 . . 3 (𝜑𝐴𝐾)
2 df-ov 6816 . . . . 5 (𝐵(𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))𝐶) = ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘⟨𝐵, 𝐶⟩)
3 xpsvsca.4 . . . . . 6 (𝜑𝐵𝑋)
4 xpsvsca.5 . . . . . 6 (𝜑𝐶𝑌)
5 eqid 2760 . . . . . . 7 (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})) = (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))
65xpsfval 16429 . . . . . 6 ((𝐵𝑋𝐶𝑌) → (𝐵(𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))𝐶) = ({𝐵} +𝑐 {𝐶}))
73, 4, 6syl2anc 696 . . . . 5 (𝜑 → (𝐵(𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))𝐶) = ({𝐵} +𝑐 {𝐶}))
82, 7syl5eqr 2808 . . . 4 (𝜑 → ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘⟨𝐵, 𝐶⟩) = ({𝐵} +𝑐 {𝐶}))
9 opelxpi 5305 . . . . . 6 ((𝐵𝑋𝐶𝑌) → ⟨𝐵, 𝐶⟩ ∈ (𝑋 × 𝑌))
103, 4, 9syl2anc 696 . . . . 5 (𝜑 → ⟨𝐵, 𝐶⟩ ∈ (𝑋 × 𝑌))
115xpsff1o2 16433 . . . . . . 7 (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)–1-1-onto→ran (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))
12 f1of 6298 . . . . . . 7 ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)–1-1-onto→ran (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})) → (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)⟶ran (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})))
1311, 12ax-mp 5 . . . . . 6 (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)⟶ran (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))
1413ffvelrni 6521 . . . . 5 (⟨𝐵, 𝐶⟩ ∈ (𝑋 × 𝑌) → ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘⟨𝐵, 𝐶⟩) ∈ ran (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})))
1510, 14syl 17 . . . 4 (𝜑 → ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘⟨𝐵, 𝐶⟩) ∈ ran (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})))
168, 15eqeltrrd 2840 . . 3 (𝜑({𝐵} +𝑐 {𝐶}) ∈ ran (𝑥𝑋, 𝑦𝑌({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 (𝜑𝑅𝑉)
21 xpssca.2 . . . . 5 (𝜑𝑆𝑊)
22 xpssca.g . . . . 5 𝐺 = (Scalar‘𝑅)
23 eqid 2760 . . . . 5 (𝐺Xs({𝑅} +𝑐 {𝑆})) = (𝐺Xs({𝑅} +𝑐 {𝑆}))
2417, 18, 19, 20, 21, 5, 22, 23xpsval 16434 . . . 4 (𝜑𝑇 = ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})) “s (𝐺Xs({𝑅} +𝑐 {𝑆}))))
2517, 18, 19, 20, 21, 5, 22, 23xpslem 16435 . . . 4 (𝜑 → ran (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})) = (Base‘(𝐺Xs({𝑅} +𝑐 {𝑆}))))
26 f1ocnv 6310 . . . . . 6 ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)–1-1-onto→ran (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})) → (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})):ran (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))–1-1-onto→(𝑋 × 𝑌))
2711, 26mp1i 13 . . . . 5 (𝜑(𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})):ran (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))–1-1-onto→(𝑋 × 𝑌))
28 f1ofo 6305 . . . . 5 ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})):ran (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))–1-1-onto→(𝑋 × 𝑌) → (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})):ran (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))–onto→(𝑋 × 𝑌))
2927, 28syl 17 . . . 4 (𝜑(𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})):ran (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))–onto→(𝑋 × 𝑌))
30 ovexd 6843 . . . 4 (𝜑 → (𝐺Xs({𝑅} +𝑐 {𝑆})) ∈ V)
31 fvex 6362 . . . . . . . 8 (Scalar‘𝑅) ∈ V
3222, 31eqeltri 2835 . . . . . . 7 𝐺 ∈ V
3332a1i 11 . . . . . 6 (⊤ → 𝐺 ∈ V)
34 ovex 6841 . . . . . . . 8 ({𝑅} +𝑐 {𝑆}) ∈ V
3534cnvex 7278 . . . . . . 7 ({𝑅} +𝑐 {𝑆}) ∈ V
3635a1i 11 . . . . . 6 (⊤ → ({𝑅} +𝑐 {𝑆}) ∈ V)
3723, 33, 36prdssca 16318 . . . . 5 (⊤ → 𝐺 = (Scalar‘(𝐺Xs({𝑅} +𝑐 {𝑆}))))
3837trud 1642 . . . 4 𝐺 = (Scalar‘(𝐺Xs({𝑅} +𝑐 {𝑆})))
39 xpsvsca.k . . . 4 𝐾 = (Base‘𝐺)
40 eqid 2760 . . . 4 ( ·𝑠 ‘(𝐺Xs({𝑅} +𝑐 {𝑆}))) = ( ·𝑠 ‘(𝐺Xs({𝑅} +𝑐 {𝑆})))
41 xpsvsca.p . . . 4 = ( ·𝑠𝑇)
4227f1ovscpbl 16388 . . . 4 ((𝜑 ∧ (𝑎𝐾𝑏 ∈ ran (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})) ∧ 𝑐 ∈ ran (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})))) → (((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘𝑏) = ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘𝑐) → ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘(𝑎( ·𝑠 ‘(𝐺Xs({𝑅} +𝑐 {𝑆})))𝑏)) = ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘(𝑎( ·𝑠 ‘(𝐺Xs({𝑅} +𝑐 {𝑆})))𝑐))))
4324, 25, 29, 30, 38, 39, 40, 41, 42imasvscaval 16400 . . 3 ((𝜑𝐴𝐾({𝐵} +𝑐 {𝐶}) ∈ ran (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))) → (𝐴 ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘({𝐵} +𝑐 {𝐶}))) = ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘(𝐴( ·𝑠 ‘(𝐺Xs({𝑅} +𝑐 {𝑆})))({𝐵} +𝑐 {𝐶}))))
441, 16, 43mpd3an23 1575 . 2 (𝜑 → (𝐴 ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘({𝐵} +𝑐 {𝐶}))) = ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘(𝐴( ·𝑠 ‘(𝐺Xs({𝑅} +𝑐 {𝑆})))({𝐵} +𝑐 {𝐶}))))
45 f1ocnvfv 6697 . . . . 5 (((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)–1-1-onto→ran (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})) ∧ ⟨𝐵, 𝐶⟩ ∈ (𝑋 × 𝑌)) → (((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘⟨𝐵, 𝐶⟩) = ({𝐵} +𝑐 {𝐶}) → ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘({𝐵} +𝑐 {𝐶})) = ⟨𝐵, 𝐶⟩))
4611, 10, 45sylancr 698 . . . 4 (𝜑 → (((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘⟨𝐵, 𝐶⟩) = ({𝐵} +𝑐 {𝐶}) → ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘({𝐵} +𝑐 {𝐶})) = ⟨𝐵, 𝐶⟩))
478, 46mpd 15 . . 3 (𝜑 → ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘({𝐵} +𝑐 {𝐶})) = ⟨𝐵, 𝐶⟩)
4847oveq2d 6829 . 2 (𝜑 → (𝐴 ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘({𝐵} +𝑐 {𝐶}))) = (𝐴 𝐵, 𝐶⟩))
49 iftrue 4236 . . . . . . . . . . . 12 (𝑘 = ∅ → if(𝑘 = ∅, 𝑅, 𝑆) = 𝑅)
5049fveq2d 6356 . . . . . . . . . . 11 (𝑘 = ∅ → ( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆)) = ( ·𝑠𝑅))
51 xpsvsca.m . . . . . . . . . . 11 · = ( ·𝑠𝑅)
5250, 51syl6eqr 2812 . . . . . . . . . 10 (𝑘 = ∅ → ( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆)) = · )
53 eqidd 2761 . . . . . . . . . 10 (𝑘 = ∅ → 𝐴 = 𝐴)
54 iftrue 4236 . . . . . . . . . 10 (𝑘 = ∅ → if(𝑘 = ∅, 𝐵, 𝐶) = 𝐵)
5552, 53, 54oveq123d 6834 . . . . . . . . 9 (𝑘 = ∅ → (𝐴( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐵, 𝐶)) = (𝐴 · 𝐵))
56 iftrue 4236 . . . . . . . . 9 (𝑘 = ∅ → if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)) = (𝐴 · 𝐵))
5755, 56eqtr4d 2797 . . . . . . . 8 (𝑘 = ∅ → (𝐴( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐵, 𝐶)) = if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)))
58 iffalse 4239 . . . . . . . . . . . 12 𝑘 = ∅ → if(𝑘 = ∅, 𝑅, 𝑆) = 𝑆)
5958fveq2d 6356 . . . . . . . . . . 11 𝑘 = ∅ → ( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆)) = ( ·𝑠𝑆))
60 xpsvsca.n . . . . . . . . . . 11 × = ( ·𝑠𝑆)
6159, 60syl6eqr 2812 . . . . . . . . . 10 𝑘 = ∅ → ( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆)) = × )
62 eqidd 2761 . . . . . . . . . 10 𝑘 = ∅ → 𝐴 = 𝐴)
63 iffalse 4239 . . . . . . . . . 10 𝑘 = ∅ → if(𝑘 = ∅, 𝐵, 𝐶) = 𝐶)
6461, 62, 63oveq123d 6834 . . . . . . . . 9 𝑘 = ∅ → (𝐴( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐵, 𝐶)) = (𝐴 × 𝐶))
65 iffalse 4239 . . . . . . . . 9 𝑘 = ∅ → if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)) = (𝐴 × 𝐶))
6664, 65eqtr4d 2797 . . . . . . . 8 𝑘 = ∅ → (𝐴( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐵, 𝐶)) = if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)))
6757, 66pm2.61i 176 . . . . . . 7 (𝐴( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐵, 𝐶)) = if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶))
6820adantr 472 . . . . . . . . . 10 ((𝜑𝑘 ∈ 2𝑜) → 𝑅𝑉)
6921adantr 472 . . . . . . . . . 10 ((𝜑𝑘 ∈ 2𝑜) → 𝑆𝑊)
70 simpr 479 . . . . . . . . . 10 ((𝜑𝑘 ∈ 2𝑜) → 𝑘 ∈ 2𝑜)
71 xpscfv 16424 . . . . . . . . . 10 ((𝑅𝑉𝑆𝑊𝑘 ∈ 2𝑜) → (({𝑅} +𝑐 {𝑆})‘𝑘) = if(𝑘 = ∅, 𝑅, 𝑆))
7268, 69, 70, 71syl3anc 1477 . . . . . . . . 9 ((𝜑𝑘 ∈ 2𝑜) → (({𝑅} +𝑐 {𝑆})‘𝑘) = if(𝑘 = ∅, 𝑅, 𝑆))
7372fveq2d 6356 . . . . . . . 8 ((𝜑𝑘 ∈ 2𝑜) → ( ·𝑠 ‘(({𝑅} +𝑐 {𝑆})‘𝑘)) = ( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆)))
74 eqidd 2761 . . . . . . . 8 ((𝜑𝑘 ∈ 2𝑜) → 𝐴 = 𝐴)
753adantr 472 . . . . . . . . 9 ((𝜑𝑘 ∈ 2𝑜) → 𝐵𝑋)
764adantr 472 . . . . . . . . 9 ((𝜑𝑘 ∈ 2𝑜) → 𝐶𝑌)
77 xpscfv 16424 . . . . . . . . 9 ((𝐵𝑋𝐶𝑌𝑘 ∈ 2𝑜) → (({𝐵} +𝑐 {𝐶})‘𝑘) = if(𝑘 = ∅, 𝐵, 𝐶))
7875, 76, 70, 77syl3anc 1477 . . . . . . . 8 ((𝜑𝑘 ∈ 2𝑜) → (({𝐵} +𝑐 {𝐶})‘𝑘) = if(𝑘 = ∅, 𝐵, 𝐶))
7973, 74, 78oveq123d 6834 . . . . . . 7 ((𝜑𝑘 ∈ 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)
8180adantr 472 . . . . . . . 8 ((𝜑𝑘 ∈ 2𝑜) → (𝐴 · 𝐵) ∈ 𝑋)
82 xpsvsca.7 . . . . . . . . 9 (𝜑 → (𝐴 × 𝐶) ∈ 𝑌)
8382adantr 472 . . . . . . . 8 ((𝜑𝑘 ∈ 2𝑜) → (𝐴 × 𝐶) ∈ 𝑌)
84 xpscfv 16424 . . . . . . . 8 (((𝐴 · 𝐵) ∈ 𝑋 ∧ (𝐴 × 𝐶) ∈ 𝑌𝑘 ∈ 2𝑜) → (({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)})‘𝑘) = if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)))
8581, 83, 70, 84syl3anc 1477 . . . . . . 7 ((𝜑𝑘 ∈ 2𝑜) → (({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)})‘𝑘) = if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)))
8667, 79, 853eqtr4a 2820 . . . . . 6 ((𝜑𝑘 ∈ 2𝑜) → (𝐴( ·𝑠 ‘(({𝑅} +𝑐 {𝑆})‘𝑘))(({𝐵} +𝑐 {𝐶})‘𝑘)) = (({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)})‘𝑘))
8786mpteq2dva 4896 . . . . 5 (𝜑 → (𝑘 ∈ 2𝑜 ↦ (𝐴( ·𝑠 ‘(({𝑅} +𝑐 {𝑆})‘𝑘))(({𝐵} +𝑐 {𝐶})‘𝑘))) = (𝑘 ∈ 2𝑜 ↦ (({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)})‘𝑘)))
88 eqid 2760 . . . . . 6 (Base‘(𝐺Xs({𝑅} +𝑐 {𝑆}))) = (Base‘(𝐺Xs({𝑅} +𝑐 {𝑆})))
8932a1i 11 . . . . . 6 (𝜑𝐺 ∈ V)
90 2on 7737 . . . . . . 7 2𝑜 ∈ On
9190a1i 11 . . . . . 6 (𝜑 → 2𝑜 ∈ On)
92 xpscfn 16421 . . . . . . 7 ((𝑅𝑉𝑆𝑊) → ({𝑅} +𝑐 {𝑆}) Fn 2𝑜)
9320, 21, 92syl2anc 696 . . . . . 6 (𝜑({𝑅} +𝑐 {𝑆}) Fn 2𝑜)
9416, 25eleqtrd 2841 . . . . . 6 (𝜑({𝐵} +𝑐 {𝐶}) ∈ (Base‘(𝐺Xs({𝑅} +𝑐 {𝑆}))))
9523, 88, 40, 39, 89, 91, 93, 1, 94prdsvscaval 16341 . . . . 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 16421 . . . . . . 7 (((𝐴 · 𝐵) ∈ 𝑋 ∧ (𝐴 × 𝐶) ∈ 𝑌) → ({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}) Fn 2𝑜)
9780, 82, 96syl2anc 696 . . . . . 6 (𝜑({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}) Fn 2𝑜)
98 dffn5 6403 . . . . . 6 (({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}) Fn 2𝑜({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}) = (𝑘 ∈ 2𝑜 ↦ (({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)})‘𝑘)))
9997, 98sylib 208 . . . . 5 (𝜑({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}) = (𝑘 ∈ 2𝑜 ↦ (({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)})‘𝑘)))
10087, 95, 993eqtr4d 2804 . . . 4 (𝜑 → (𝐴( ·𝑠 ‘(𝐺Xs({𝑅} +𝑐 {𝑆})))({𝐵} +𝑐 {𝐶})) = ({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}))
101100fveq2d 6356 . . 3 (𝜑 → ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘(𝐴( ·𝑠 ‘(𝐺Xs({𝑅} +𝑐 {𝑆})))({𝐵} +𝑐 {𝐶}))) = ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)})))
102 df-ov 6816 . . . . 5 ((𝐴 · 𝐵)(𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))(𝐴 × 𝐶)) = ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩)
1035xpsfval 16429 . . . . . 6 (((𝐴 · 𝐵) ∈ 𝑋 ∧ (𝐴 × 𝐶) ∈ 𝑌) → ((𝐴 · 𝐵)(𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))(𝐴 × 𝐶)) = ({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}))
10480, 82, 103syl2anc 696 . . . . 5 (𝜑 → ((𝐴 · 𝐵)(𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))(𝐴 × 𝐶)) = ({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}))
105102, 104syl5eqr 2808 . . . 4 (𝜑 → ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩) = ({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}))
106 opelxpi 5305 . . . . . 6 (((𝐴 · 𝐵) ∈ 𝑋 ∧ (𝐴 × 𝐶) ∈ 𝑌) → ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩ ∈ (𝑋 × 𝑌))
10780, 82, 106syl2anc 696 . . . . 5 (𝜑 → ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩ ∈ (𝑋 × 𝑌))
108 f1ocnvfv 6697 . . . . 5 (((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)–1-1-onto→ran (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})) ∧ ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩ ∈ (𝑋 × 𝑌)) → (((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩) = ({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}) → ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)})) = ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩))
10911, 107, 108sylancr 698 . . . 4 (𝜑 → (((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩) = ({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}) → ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)})) = ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩))
110105, 109mpd 15 . . 3 (𝜑 → ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)})) = ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩)
111101, 110eqtrd 2794 . 2 (𝜑 → ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘(𝐴( ·𝑠 ‘(𝐺Xs({𝑅} +𝑐 {𝑆})))({𝐵} +𝑐 {𝐶}))) = ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩)
11244, 48, 1113eqtr3d 2802 1 (𝜑 → (𝐴 𝐵, 𝐶⟩) = ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1632  wtru 1633  wcel 2139  Vcvv 3340  c0 4058  ifcif 4230  {csn 4321  cop 4327  cmpt 4881   × cxp 5264  ccnv 5265  ran crn 5267  Oncon0 5884   Fn wfn 6044  wf 6045  ontowfo 6047  1-1-ontowf1o 6048  cfv 6049  (class class class)co 6813  cmpt2 6815  2𝑜c2o 7723   +𝑐 ccda 9181  Basecbs 16059  Scalarcsca 16146   ·𝑠 cvsca 16147  Xscprds 16308   ×s cxps 16368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-map 8025  df-ixp 8075  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-sup 8513  df-inf 8514  df-cda 9182  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-n0 11485  df-z 11570  df-dec 11686  df-uz 11880  df-fz 12520  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-plusg 16156  df-mulr 16157  df-sca 16159  df-vsca 16160  df-ip 16161  df-tset 16162  df-ple 16163  df-ds 16166  df-hom 16168  df-cco 16169  df-prds 16310  df-imas 16370  df-xps 16372
This theorem is referenced by: (None)
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