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Theorem xpsvsca 16447
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 6796 . . . . 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 2771 . . . . . . 7 (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})) = (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))
65xpsfval 16435 . . . . . 6 ((𝐵𝑋𝐶𝑌) → (𝐵(𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))𝐶) = ({𝐵} +𝑐 {𝐶}))
73, 4, 6syl2anc 573 . . . . 5 (𝜑 → (𝐵(𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))𝐶) = ({𝐵} +𝑐 {𝐶}))
82, 7syl5eqr 2819 . . . 4 (𝜑 → ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘⟨𝐵, 𝐶⟩) = ({𝐵} +𝑐 {𝐶}))
9 opelxpi 5288 . . . . . 6 ((𝐵𝑋𝐶𝑌) → ⟨𝐵, 𝐶⟩ ∈ (𝑋 × 𝑌))
103, 4, 9syl2anc 573 . . . . 5 (𝜑 → ⟨𝐵, 𝐶⟩ ∈ (𝑋 × 𝑌))
115xpsff1o2 16439 . . . . . . 7 (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)–1-1-onto→ran (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))
12 f1of 6278 . . . . . . 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 6501 . . . . 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 2851 . . 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 2771 . . . . 5 (𝐺Xs({𝑅} +𝑐 {𝑆})) = (𝐺Xs({𝑅} +𝑐 {𝑆}))
2417, 18, 19, 20, 21, 5, 22, 23xpsval 16440 . . . 4 (𝜑𝑇 = ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})) “s (𝐺Xs({𝑅} +𝑐 {𝑆}))))
2517, 18, 19, 20, 21, 5, 22, 23xpslem 16441 . . . 4 (𝜑 → ran (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})) = (Base‘(𝐺Xs({𝑅} +𝑐 {𝑆}))))
26 f1ocnv 6290 . . . . . 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 6285 . . . . 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 6825 . . . 4 (𝜑 → (𝐺Xs({𝑅} +𝑐 {𝑆})) ∈ V)
31 fvex 6342 . . . . . . . 8 (Scalar‘𝑅) ∈ V
3222, 31eqeltri 2846 . . . . . . 7 𝐺 ∈ V
3332a1i 11 . . . . . 6 (⊤ → 𝐺 ∈ V)
34 ovex 6823 . . . . . . . 8 ({𝑅} +𝑐 {𝑆}) ∈ V
3534cnvex 7260 . . . . . . 7 ({𝑅} +𝑐 {𝑆}) ∈ V
3635a1i 11 . . . . . 6 (⊤ → ({𝑅} +𝑐 {𝑆}) ∈ V)
3723, 33, 36prdssca 16324 . . . . 5 (⊤ → 𝐺 = (Scalar‘(𝐺Xs({𝑅} +𝑐 {𝑆}))))
3837trud 1641 . . . 4 𝐺 = (Scalar‘(𝐺Xs({𝑅} +𝑐 {𝑆})))
39 xpsvsca.k . . . 4 𝐾 = (Base‘𝐺)
40 eqid 2771 . . . 4 ( ·𝑠 ‘(𝐺Xs({𝑅} +𝑐 {𝑆}))) = ( ·𝑠 ‘(𝐺Xs({𝑅} +𝑐 {𝑆})))
41 xpsvsca.p . . . 4 = ( ·𝑠𝑇)
4227f1ovscpbl 16394 . . . 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 16406 . . 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 1574 . 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 6677 . . . . 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 575 . . . 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 6809 . 2 (𝜑 → (𝐴 ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘({𝐵} +𝑐 {𝐶}))) = (𝐴 𝐵, 𝐶⟩))
49 iftrue 4231 . . . . . . . . . . . 12 (𝑘 = ∅ → if(𝑘 = ∅, 𝑅, 𝑆) = 𝑅)
5049fveq2d 6336 . . . . . . . . . . 11 (𝑘 = ∅ → ( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆)) = ( ·𝑠𝑅))
51 xpsvsca.m . . . . . . . . . . 11 · = ( ·𝑠𝑅)
5250, 51syl6eqr 2823 . . . . . . . . . 10 (𝑘 = ∅ → ( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆)) = · )
53 eqidd 2772 . . . . . . . . . 10 (𝑘 = ∅ → 𝐴 = 𝐴)
54 iftrue 4231 . . . . . . . . . 10 (𝑘 = ∅ → if(𝑘 = ∅, 𝐵, 𝐶) = 𝐵)
5552, 53, 54oveq123d 6814 . . . . . . . . 9 (𝑘 = ∅ → (𝐴( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐵, 𝐶)) = (𝐴 · 𝐵))
56 iftrue 4231 . . . . . . . . 9 (𝑘 = ∅ → if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)) = (𝐴 · 𝐵))
5755, 56eqtr4d 2808 . . . . . . . 8 (𝑘 = ∅ → (𝐴( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐵, 𝐶)) = if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)))
58 iffalse 4234 . . . . . . . . . . . 12 𝑘 = ∅ → if(𝑘 = ∅, 𝑅, 𝑆) = 𝑆)
5958fveq2d 6336 . . . . . . . . . . 11 𝑘 = ∅ → ( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆)) = ( ·𝑠𝑆))
60 xpsvsca.n . . . . . . . . . . 11 × = ( ·𝑠𝑆)
6159, 60syl6eqr 2823 . . . . . . . . . 10 𝑘 = ∅ → ( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆)) = × )
62 eqidd 2772 . . . . . . . . . 10 𝑘 = ∅ → 𝐴 = 𝐴)
63 iffalse 4234 . . . . . . . . . 10 𝑘 = ∅ → if(𝑘 = ∅, 𝐵, 𝐶) = 𝐶)
6461, 62, 63oveq123d 6814 . . . . . . . . 9 𝑘 = ∅ → (𝐴( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐵, 𝐶)) = (𝐴 × 𝐶))
65 iffalse 4234 . . . . . . . . 9 𝑘 = ∅ → if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)) = (𝐴 × 𝐶))
6664, 65eqtr4d 2808 . . . . . . . 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 466 . . . . . . . . . 10 ((𝜑𝑘 ∈ 2𝑜) → 𝑅𝑉)
6921adantr 466 . . . . . . . . . 10 ((𝜑𝑘 ∈ 2𝑜) → 𝑆𝑊)
70 simpr 471 . . . . . . . . . 10 ((𝜑𝑘 ∈ 2𝑜) → 𝑘 ∈ 2𝑜)
71 xpscfv 16430 . . . . . . . . . 10 ((𝑅𝑉𝑆𝑊𝑘 ∈ 2𝑜) → (({𝑅} +𝑐 {𝑆})‘𝑘) = if(𝑘 = ∅, 𝑅, 𝑆))
7268, 69, 70, 71syl3anc 1476 . . . . . . . . 9 ((𝜑𝑘 ∈ 2𝑜) → (({𝑅} +𝑐 {𝑆})‘𝑘) = if(𝑘 = ∅, 𝑅, 𝑆))
7372fveq2d 6336 . . . . . . . 8 ((𝜑𝑘 ∈ 2𝑜) → ( ·𝑠 ‘(({𝑅} +𝑐 {𝑆})‘𝑘)) = ( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆)))
74 eqidd 2772 . . . . . . . 8 ((𝜑𝑘 ∈ 2𝑜) → 𝐴 = 𝐴)
753adantr 466 . . . . . . . . 9 ((𝜑𝑘 ∈ 2𝑜) → 𝐵𝑋)
764adantr 466 . . . . . . . . 9 ((𝜑𝑘 ∈ 2𝑜) → 𝐶𝑌)
77 xpscfv 16430 . . . . . . . . 9 ((𝐵𝑋𝐶𝑌𝑘 ∈ 2𝑜) → (({𝐵} +𝑐 {𝐶})‘𝑘) = if(𝑘 = ∅, 𝐵, 𝐶))
7875, 76, 70, 77syl3anc 1476 . . . . . . . 8 ((𝜑𝑘 ∈ 2𝑜) → (({𝐵} +𝑐 {𝐶})‘𝑘) = if(𝑘 = ∅, 𝐵, 𝐶))
7973, 74, 78oveq123d 6814 . . . . . . 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 466 . . . . . . . 8 ((𝜑𝑘 ∈ 2𝑜) → (𝐴 · 𝐵) ∈ 𝑋)
82 xpsvsca.7 . . . . . . . . 9 (𝜑 → (𝐴 × 𝐶) ∈ 𝑌)
8382adantr 466 . . . . . . . 8 ((𝜑𝑘 ∈ 2𝑜) → (𝐴 × 𝐶) ∈ 𝑌)
84 xpscfv 16430 . . . . . . . 8 (((𝐴 · 𝐵) ∈ 𝑋 ∧ (𝐴 × 𝐶) ∈ 𝑌𝑘 ∈ 2𝑜) → (({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)})‘𝑘) = if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)))
8581, 83, 70, 84syl3anc 1476 . . . . . . 7 ((𝜑𝑘 ∈ 2𝑜) → (({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)})‘𝑘) = if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)))
8667, 79, 853eqtr4a 2831 . . . . . 6 ((𝜑𝑘 ∈ 2𝑜) → (𝐴( ·𝑠 ‘(({𝑅} +𝑐 {𝑆})‘𝑘))(({𝐵} +𝑐 {𝐶})‘𝑘)) = (({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)})‘𝑘))
8786mpteq2dva 4878 . . . . 5 (𝜑 → (𝑘 ∈ 2𝑜 ↦ (𝐴( ·𝑠 ‘(({𝑅} +𝑐 {𝑆})‘𝑘))(({𝐵} +𝑐 {𝐶})‘𝑘))) = (𝑘 ∈ 2𝑜 ↦ (({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)})‘𝑘)))
88 eqid 2771 . . . . . 6 (Base‘(𝐺Xs({𝑅} +𝑐 {𝑆}))) = (Base‘(𝐺Xs({𝑅} +𝑐 {𝑆})))
8932a1i 11 . . . . . 6 (𝜑𝐺 ∈ V)
90 2on 7722 . . . . . . 7 2𝑜 ∈ On
9190a1i 11 . . . . . 6 (𝜑 → 2𝑜 ∈ On)
92 xpscfn 16427 . . . . . . 7 ((𝑅𝑉𝑆𝑊) → ({𝑅} +𝑐 {𝑆}) Fn 2𝑜)
9320, 21, 92syl2anc 573 . . . . . 6 (𝜑({𝑅} +𝑐 {𝑆}) Fn 2𝑜)
9416, 25eleqtrd 2852 . . . . . 6 (𝜑({𝐵} +𝑐 {𝐶}) ∈ (Base‘(𝐺Xs({𝑅} +𝑐 {𝑆}))))
9523, 88, 40, 39, 89, 91, 93, 1, 94prdsvscaval 16347 . . . . 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 16427 . . . . . . 7 (((𝐴 · 𝐵) ∈ 𝑋 ∧ (𝐴 × 𝐶) ∈ 𝑌) → ({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}) Fn 2𝑜)
9780, 82, 96syl2anc 573 . . . . . 6 (𝜑({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}) Fn 2𝑜)
98 dffn5 6383 . . . . . 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 2815 . . . 4 (𝜑 → (𝐴( ·𝑠 ‘(𝐺Xs({𝑅} +𝑐 {𝑆})))({𝐵} +𝑐 {𝐶})) = ({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}))
101100fveq2d 6336 . . 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 6796 . . . . 5 ((𝐴 · 𝐵)(𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))(𝐴 × 𝐶)) = ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩)
1035xpsfval 16435 . . . . . 6 (((𝐴 · 𝐵) ∈ 𝑋 ∧ (𝐴 × 𝐶) ∈ 𝑌) → ((𝐴 · 𝐵)(𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))(𝐴 × 𝐶)) = ({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}))
10480, 82, 103syl2anc 573 . . . . 5 (𝜑 → ((𝐴 · 𝐵)(𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))(𝐴 × 𝐶)) = ({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}))
105102, 104syl5eqr 2819 . . . 4 (𝜑 → ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩) = ({(𝐴 · 𝐵)} +𝑐 {(𝐴 × 𝐶)}))
106 opelxpi 5288 . . . . . 6 (((𝐴 · 𝐵) ∈ 𝑋 ∧ (𝐴 × 𝐶) ∈ 𝑌) → ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩ ∈ (𝑋 × 𝑌))
10780, 82, 106syl2anc 573 . . . . 5 (𝜑 → ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩ ∈ (𝑋 × 𝑌))
108 f1ocnvfv 6677 . . . . 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 575 . . . 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 2805 . 2 (𝜑 → ((𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))‘(𝐴( ·𝑠 ‘(𝐺Xs({𝑅} +𝑐 {𝑆})))({𝐵} +𝑐 {𝐶}))) = ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩)
11244, 48, 1113eqtr3d 2813 1 (𝜑 → (𝐴 𝐵, 𝐶⟩) = ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1631  wtru 1632  wcel 2145  Vcvv 3351  c0 4063  ifcif 4225  {csn 4316  cop 4322  cmpt 4863   × cxp 5247  ccnv 5248  ran crn 5250  Oncon0 5866   Fn wfn 6026  wf 6027  ontowfo 6029  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6793  cmpt2 6795  2𝑜c2o 7707   +𝑐 ccda 9191  Basecbs 16064  Scalarcsca 16152   ·𝑠 cvsca 16153  Xscprds 16314   ×s cxps 16374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-map 8011  df-ixp 8063  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-sup 8504  df-inf 8505  df-cda 9192  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-2 11281  df-3 11282  df-4 11283  df-5 11284  df-6 11285  df-7 11286  df-8 11287  df-9 11288  df-n0 11495  df-z 11580  df-dec 11696  df-uz 11889  df-fz 12534  df-struct 16066  df-ndx 16067  df-slot 16068  df-base 16070  df-plusg 16162  df-mulr 16163  df-sca 16165  df-vsca 16166  df-ip 16167  df-tset 16168  df-ple 16169  df-ds 16172  df-hom 16174  df-cco 16175  df-prds 16316  df-imas 16376  df-xps 16378
This theorem is referenced by: (None)
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