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Theorem xpsvsca 16155
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 6608 . . . . 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 2626 . . . . . . 7 (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})) = (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))
65xpsfval 16143 . . . . . 6 ((𝐵𝑋𝐶𝑌) → (𝐵(𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))𝐶) = ({𝐵} +𝑐 {𝐶}))
73, 4, 6syl2anc 692 . . . . 5 (𝜑 → (𝐵(𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))𝐶) = ({𝐵} +𝑐 {𝐶}))
82, 7syl5eqr 2674 . . . 4 (𝜑 → ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘⟨𝐵, 𝐶⟩) = ({𝐵} +𝑐 {𝐶}))
9 opelxpi 5113 . . . . . 6 ((𝐵𝑋𝐶𝑌) → ⟨𝐵, 𝐶⟩ ∈ (𝑋 × 𝑌))
103, 4, 9syl2anc 692 . . . . 5 (𝜑 → ⟨𝐵, 𝐶⟩ ∈ (𝑋 × 𝑌))
115xpsff1o2 16147 . . . . . . 7 (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)–1-1-onto→ran (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))
12 f1of 6096 . . . . . . 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 6315 . . . . 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 2705 . . 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 2626 . . . . 5 (𝐺Xs({𝑅} +𝑐 {𝑆})) = (𝐺Xs({𝑅} +𝑐 {𝑆}))
2417, 18, 19, 20, 21, 5, 22, 23xpsval 16148 . . . 4 (𝜑𝑇 = ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})) “s (𝐺Xs({𝑅} +𝑐 {𝑆}))))
2517, 18, 19, 20, 21, 5, 22, 23xpslem 16149 . . . 4 (𝜑 → ran (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})) = (Base‘(𝐺Xs({𝑅} +𝑐 {𝑆}))))
26 f1ocnv 6108 . . . . . 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 6103 . . . . 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 ovex 6633 . . . . 5 (𝐺Xs({𝑅} +𝑐 {𝑆})) ∈ V
3130a1i 11 . . . 4 (𝜑 → (𝐺Xs({𝑅} +𝑐 {𝑆})) ∈ V)
32 fvex 6160 . . . . . . . 8 (Scalar‘𝑅) ∈ V
3322, 32eqeltri 2700 . . . . . . 7 𝐺 ∈ V
3433a1i 11 . . . . . 6 (⊤ → 𝐺 ∈ V)
35 ovex 6633 . . . . . . . 8 ({𝑅} +𝑐 {𝑆}) ∈ V
3635cnvex 7063 . . . . . . 7 ({𝑅} +𝑐 {𝑆}) ∈ V
3736a1i 11 . . . . . 6 (⊤ → ({𝑅} +𝑐 {𝑆}) ∈ V)
3823, 34, 37prdssca 16032 . . . . 5 (⊤ → 𝐺 = (Scalar‘(𝐺Xs({𝑅} +𝑐 {𝑆}))))
3938trud 1490 . . . 4 𝐺 = (Scalar‘(𝐺Xs({𝑅} +𝑐 {𝑆})))
40 xpsvsca.k . . . 4 𝐾 = (Base‘𝐺)
41 eqid 2626 . . . 4 ( ·𝑠 ‘(𝐺Xs({𝑅} +𝑐 {𝑆}))) = ( ·𝑠 ‘(𝐺Xs({𝑅} +𝑐 {𝑆})))
42 xpsvsca.p . . . 4 = ( ·𝑠𝑇)
4327f1ovscpbl 16102 . . . 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))))
4424, 25, 29, 31, 39, 40, 41, 42, 43imasvscaval 16114 . . 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}))))
451, 16, 44mpd3an23 1423 . 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}))))
46 f1ocnvfv 6489 . . . . 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⟩))
4711, 10, 46sylancr 694 . . . 4 (𝜑 → (((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘⟨𝐵, 𝐶⟩) = ({𝐵} +𝑐 {𝐶}) → ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘({𝐵} +𝑐 {𝐶})) = ⟨𝐵, 𝐶⟩))
488, 47mpd 15 . . 3 (𝜑 → ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘({𝐵} +𝑐 {𝐶})) = ⟨𝐵, 𝐶⟩)
4948oveq2d 6621 . 2 (𝜑 → (𝐴 ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘({𝐵} +𝑐 {𝐶}))) = (𝐴 𝐵, 𝐶⟩))
50 iftrue 4069 . . . . . . . . . . . 12 (𝑘 = ∅ → if(𝑘 = ∅, 𝑅, 𝑆) = 𝑅)
5150fveq2d 6154 . . . . . . . . . . 11 (𝑘 = ∅ → ( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆)) = ( ·𝑠𝑅))
52 xpsvsca.m . . . . . . . . . . 11 · = ( ·𝑠𝑅)
5351, 52syl6eqr 2678 . . . . . . . . . 10 (𝑘 = ∅ → ( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆)) = · )
54 eqidd 2627 . . . . . . . . . 10 (𝑘 = ∅ → 𝐴 = 𝐴)
55 iftrue 4069 . . . . . . . . . 10 (𝑘 = ∅ → if(𝑘 = ∅, 𝐵, 𝐶) = 𝐵)
5653, 54, 55oveq123d 6626 . . . . . . . . 9 (𝑘 = ∅ → (𝐴( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐵, 𝐶)) = (𝐴 · 𝐵))
57 iftrue 4069 . . . . . . . . 9 (𝑘 = ∅ → if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)) = (𝐴 · 𝐵))
5856, 57eqtr4d 2663 . . . . . . . 8 (𝑘 = ∅ → (𝐴( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐵, 𝐶)) = if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)))
59 iffalse 4072 . . . . . . . . . . . 12 𝑘 = ∅ → if(𝑘 = ∅, 𝑅, 𝑆) = 𝑆)
6059fveq2d 6154 . . . . . . . . . . 11 𝑘 = ∅ → ( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆)) = ( ·𝑠𝑆))
61 xpsvsca.n . . . . . . . . . . 11 × = ( ·𝑠𝑆)
6260, 61syl6eqr 2678 . . . . . . . . . 10 𝑘 = ∅ → ( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆)) = × )
63 eqidd 2627 . . . . . . . . . 10 𝑘 = ∅ → 𝐴 = 𝐴)
64 iffalse 4072 . . . . . . . . . 10 𝑘 = ∅ → if(𝑘 = ∅, 𝐵, 𝐶) = 𝐶)
6562, 63, 64oveq123d 6626 . . . . . . . . 9 𝑘 = ∅ → (𝐴( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐵, 𝐶)) = (𝐴 × 𝐶))
66 iffalse 4072 . . . . . . . . 9 𝑘 = ∅ → if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)) = (𝐴 × 𝐶))
6765, 66eqtr4d 2663 . . . . . . . 8 𝑘 = ∅ → (𝐴( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐵, 𝐶)) = if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)))
6858, 67pm2.61i 176 . . . . . . 7 (𝐴( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐵, 𝐶)) = if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶))
6920adantr 481 . . . . . . . . . 10 ((𝜑𝑘 ∈ 2𝑜) → 𝑅𝑉)
7021adantr 481 . . . . . . . . . 10 ((𝜑𝑘 ∈ 2𝑜) → 𝑆𝑊)
71 simpr 477 . . . . . . . . . 10 ((𝜑𝑘 ∈ 2𝑜) → 𝑘 ∈ 2𝑜)
72 xpscfv 16138 . . . . . . . . . 10 ((𝑅𝑉𝑆𝑊𝑘 ∈ 2𝑜) → (({𝑅} +𝑐 {𝑆})‘𝑘) = if(𝑘 = ∅, 𝑅, 𝑆))
7369, 70, 71, 72syl3anc 1323 . . . . . . . . 9 ((𝜑𝑘 ∈ 2𝑜) → (({𝑅} +𝑐 {𝑆})‘𝑘) = if(𝑘 = ∅, 𝑅, 𝑆))
7473fveq2d 6154 . . . . . . . 8 ((𝜑𝑘 ∈ 2𝑜) → ( ·𝑠 ‘(({𝑅} +𝑐 {𝑆})‘𝑘)) = ( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆)))
75 eqidd 2627 . . . . . . . 8 ((𝜑𝑘 ∈ 2𝑜) → 𝐴 = 𝐴)
763adantr 481 . . . . . . . . 9 ((𝜑𝑘 ∈ 2𝑜) → 𝐵𝑋)
774adantr 481 . . . . . . . . 9 ((𝜑𝑘 ∈ 2𝑜) → 𝐶𝑌)
78 xpscfv 16138 . . . . . . . . 9 ((𝐵𝑋𝐶𝑌𝑘 ∈ 2𝑜) → (({𝐵} +𝑐 {𝐶})‘𝑘) = if(𝑘 = ∅, 𝐵, 𝐶))
7976, 77, 71, 78syl3anc 1323 . . . . . . . 8 ((𝜑𝑘 ∈ 2𝑜) → (({𝐵} +𝑐 {𝐶})‘𝑘) = if(𝑘 = ∅, 𝐵, 𝐶))
8074, 75, 79oveq123d 6626 . . . . . . 7 ((𝜑𝑘 ∈ 2𝑜) → (𝐴( ·𝑠 ‘(({𝑅} +𝑐 {𝑆})‘𝑘))(({𝐵} +𝑐 {𝐶})‘𝑘)) = (𝐴( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐵, 𝐶)))
81 xpsvsca.6 . . . . . . . . 9 (𝜑 → (𝐴 · 𝐵) ∈ 𝑋)
8281adantr 481 . . . . . . . 8 ((𝜑𝑘 ∈ 2𝑜) → (𝐴 · 𝐵) ∈ 𝑋)
83 xpsvsca.7 . . . . . . . . 9 (𝜑 → (𝐴 × 𝐶) ∈ 𝑌)
8483adantr 481 . . . . . . . 8 ((𝜑𝑘 ∈ 2𝑜) → (𝐴 × 𝐶) ∈ 𝑌)
85 xpscfv 16138 . . . . . . . 8 (((𝐴 · 𝐵) ∈ 𝑋 ∧ (𝐴 × 𝐶) ∈ 𝑌𝑘 ∈ 2𝑜) → (({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)})‘𝑘) = if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)))
8682, 84, 71, 85syl3anc 1323 . . . . . . 7 ((𝜑𝑘 ∈ 2𝑜) → (({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)})‘𝑘) = if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)))
8768, 80, 863eqtr4a 2686 . . . . . 6 ((𝜑𝑘 ∈ 2𝑜) → (𝐴( ·𝑠 ‘(({𝑅} +𝑐 {𝑆})‘𝑘))(({𝐵} +𝑐 {𝐶})‘𝑘)) = (({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)})‘𝑘))
8887mpteq2dva 4709 . . . . 5 (𝜑 → (𝑘 ∈ 2𝑜 ↦ (𝐴( ·𝑠 ‘(({𝑅} +𝑐 {𝑆})‘𝑘))(({𝐵} +𝑐 {𝐶})‘𝑘))) = (𝑘 ∈ 2𝑜 ↦ (({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)})‘𝑘)))
89 eqid 2626 . . . . . 6 (Base‘(𝐺Xs({𝑅} +𝑐 {𝑆}))) = (Base‘(𝐺Xs({𝑅} +𝑐 {𝑆})))
9033a1i 11 . . . . . 6 (𝜑𝐺 ∈ V)
91 2on 7514 . . . . . . 7 2𝑜 ∈ On
9291a1i 11 . . . . . 6 (𝜑 → 2𝑜 ∈ On)
93 xpscfn 16135 . . . . . . 7 ((𝑅𝑉𝑆𝑊) → ({𝑅} +𝑐 {𝑆}) Fn 2𝑜)
9420, 21, 93syl2anc 692 . . . . . 6 (𝜑({𝑅} +𝑐 {𝑆}) Fn 2𝑜)
9516, 25eleqtrd 2706 . . . . . 6 (𝜑({𝐵} +𝑐 {𝐶}) ∈ (Base‘(𝐺Xs({𝑅} +𝑐 {𝑆}))))
9623, 89, 41, 40, 90, 92, 94, 1, 95prdsvscaval 16055 . . . . 5 (𝜑 → (𝐴( ·𝑠 ‘(𝐺Xs({𝑅} +𝑐 {𝑆})))({𝐵} +𝑐 {𝐶})) = (𝑘 ∈ 2𝑜 ↦ (𝐴( ·𝑠 ‘(({𝑅} +𝑐 {𝑆})‘𝑘))(({𝐵} +𝑐 {𝐶})‘𝑘))))
97 xpscfn 16135 . . . . . . 7 (((𝐴 · 𝐵) ∈ 𝑋 ∧ (𝐴 × 𝐶) ∈ 𝑌) → ({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}) Fn 2𝑜)
9881, 83, 97syl2anc 692 . . . . . 6 (𝜑({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}) Fn 2𝑜)
99 dffn5 6199 . . . . . 6 (({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}) Fn 2𝑜({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}) = (𝑘 ∈ 2𝑜 ↦ (({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)})‘𝑘)))
10098, 99sylib 208 . . . . 5 (𝜑({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}) = (𝑘 ∈ 2𝑜 ↦ (({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)})‘𝑘)))
10188, 96, 1003eqtr4d 2670 . . . 4 (𝜑 → (𝐴( ·𝑠 ‘(𝐺Xs({𝑅} +𝑐 {𝑆})))({𝐵} +𝑐 {𝐶})) = ({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}))
102101fveq2d 6154 . . 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)})))
103 df-ov 6608 . . . . 5 ((𝐴 · 𝐵)(𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))(𝐴 × 𝐶)) = ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩)
1045xpsfval 16143 . . . . . 6 (((𝐴 · 𝐵) ∈ 𝑋 ∧ (𝐴 × 𝐶) ∈ 𝑌) → ((𝐴 · 𝐵)(𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))(𝐴 × 𝐶)) = ({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}))
10581, 83, 104syl2anc 692 . . . . 5 (𝜑 → ((𝐴 · 𝐵)(𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))(𝐴 × 𝐶)) = ({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}))
106103, 105syl5eqr 2674 . . . 4 (𝜑 → ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩) = ({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}))
107 opelxpi 5113 . . . . . 6 (((𝐴 · 𝐵) ∈ 𝑋 ∧ (𝐴 × 𝐶) ∈ 𝑌) → ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩ ∈ (𝑋 × 𝑌))
10881, 83, 107syl2anc 692 . . . . 5 (𝜑 → ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩ ∈ (𝑋 × 𝑌))
109 f1ocnvfv 6489 . . . . 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)⟩))
11011, 108, 109sylancr 694 . . . 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)⟩))
111106, 110mpd 15 . . 3 (𝜑 → ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)})) = ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩)
112102, 111eqtrd 2660 . 2 (𝜑 → ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘(𝐴( ·𝑠 ‘(𝐺Xs({𝑅} +𝑐 {𝑆})))({𝐵} +𝑐 {𝐶}))) = ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩)
11345, 49, 1123eqtr3d 2668 1 (𝜑 → (𝐴 𝐵, 𝐶⟩) = ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wtru 1481  wcel 1992  Vcvv 3191  c0 3896  ifcif 4063  {csn 4153  cop 4159  cmpt 4678   × cxp 5077  ccnv 5078  ran crn 5080  Oncon0 5685   Fn wfn 5845  wf 5846  ontowfo 5848  1-1-ontowf1o 5849  cfv 5850  (class class class)co 6605  cmpt2 6607  2𝑜c2o 7500   +𝑐 ccda 8934  Basecbs 15776  Scalarcsca 15860   ·𝑠 cvsca 15861  Xscprds 16022   ×s cxps 16082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-2o 7507  df-oadd 7510  df-er 7688  df-map 7805  df-ixp 7854  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-sup 8293  df-inf 8294  df-cda 8935  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-3 11025  df-4 11026  df-5 11027  df-6 11028  df-7 11029  df-8 11030  df-9 11031  df-n0 11238  df-z 11323  df-dec 11438  df-uz 11632  df-fz 12266  df-struct 15778  df-ndx 15779  df-slot 15780  df-base 15781  df-plusg 15870  df-mulr 15871  df-sca 15873  df-vsca 15874  df-ip 15875  df-tset 15876  df-ple 15877  df-ds 15880  df-hom 15882  df-cco 15883  df-prds 16024  df-imas 16084  df-xps 16086
This theorem is referenced by: (None)
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