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Mirrors > Home > MPE Home > Th. List > xpssca | Structured version Visualization version GIF version |
Description: Value of the scalar field of a binary structure product. For concreteness, we choose the scalar field to match the left argument, but in most cases where this slot is meaningful both factors will have the same scalar field, so that it doesn't matter which factor is chosen. (Contributed by Mario Carneiro, 15-Aug-2015.) |
Ref | Expression |
---|---|
xpssca.t | ⊢ 𝑇 = (𝑅 ×s 𝑆) |
xpssca.g | ⊢ 𝐺 = (Scalar‘𝑅) |
xpssca.1 | ⊢ (𝜑 → 𝑅 ∈ 𝑉) |
xpssca.2 | ⊢ (𝜑 → 𝑆 ∈ 𝑊) |
Ref | Expression |
---|---|
xpssca | ⊢ (𝜑 → 𝐺 = (Scalar‘𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2820 | . . 3 ⊢ (𝐺Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}) = (𝐺Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}) | |
2 | xpssca.g | . . . . 5 ⊢ 𝐺 = (Scalar‘𝑅) | |
3 | 2 | fvexi 6677 | . . . 4 ⊢ 𝐺 ∈ V |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐺 ∈ V) |
5 | prex 5326 | . . . 4 ⊢ {〈∅, 𝑅〉, 〈1o, 𝑆〉} ∈ V | |
6 | 5 | a1i 11 | . . 3 ⊢ (𝜑 → {〈∅, 𝑅〉, 〈1o, 𝑆〉} ∈ V) |
7 | 1, 4, 6 | prdssca 16724 | . 2 ⊢ (𝜑 → 𝐺 = (Scalar‘(𝐺Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}))) |
8 | xpssca.t | . . . 4 ⊢ 𝑇 = (𝑅 ×s 𝑆) | |
9 | eqid 2820 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
10 | eqid 2820 | . . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
11 | xpssca.1 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑉) | |
12 | xpssca.2 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑊) | |
13 | eqid 2820 | . . . 4 ⊢ (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) = (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) | |
14 | 8, 9, 10, 11, 12, 13, 2, 1 | xpsval 16838 | . . 3 ⊢ (𝜑 → 𝑇 = (◡(𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) “s (𝐺Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}))) |
15 | 8, 9, 10, 11, 12, 13, 2, 1 | xpsrnbas 16839 | . . 3 ⊢ (𝜑 → ran (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) = (Base‘(𝐺Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}))) |
16 | 13 | xpsff1o2 16837 | . . . . 5 ⊢ (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):((Base‘𝑅) × (Base‘𝑆))–1-1-onto→ran (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) |
17 | f1ocnv 6620 | . . . . 5 ⊢ ((𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):((Base‘𝑅) × (Base‘𝑆))–1-1-onto→ran (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) → ◡(𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):ran (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉})–1-1-onto→((Base‘𝑅) × (Base‘𝑆))) | |
18 | 16, 17 | mp1i 13 | . . . 4 ⊢ (𝜑 → ◡(𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):ran (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉})–1-1-onto→((Base‘𝑅) × (Base‘𝑆))) |
19 | f1ofo 6615 | . . . 4 ⊢ (◡(𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):ran (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉})–1-1-onto→((Base‘𝑅) × (Base‘𝑆)) → ◡(𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):ran (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉})–onto→((Base‘𝑅) × (Base‘𝑆))) | |
20 | 18, 19 | syl 17 | . . 3 ⊢ (𝜑 → ◡(𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):ran (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉})–onto→((Base‘𝑅) × (Base‘𝑆))) |
21 | ovexd 7184 | . . 3 ⊢ (𝜑 → (𝐺Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}) ∈ V) | |
22 | eqid 2820 | . . 3 ⊢ (Scalar‘(𝐺Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉})) = (Scalar‘(𝐺Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉})) | |
23 | 14, 15, 20, 21, 22 | imassca 16787 | . 2 ⊢ (𝜑 → (Scalar‘(𝐺Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉})) = (Scalar‘𝑇)) |
24 | 7, 23 | eqtrd 2855 | 1 ⊢ (𝜑 → 𝐺 = (Scalar‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 Vcvv 3491 ∅c0 4284 {cpr 4562 〈cop 4566 × cxp 5546 ◡ccnv 5547 ran crn 5549 –onto→wfo 6346 –1-1-onto→wf1o 6347 ‘cfv 6348 (class class class)co 7149 ∈ cmpo 7151 1oc1o 8088 Basecbs 16478 Scalarcsca 16563 Xscprds 16714 ×s cxps 16774 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-2o 8096 df-oadd 8099 df-er 8282 df-map 8401 df-ixp 8455 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-sup 8899 df-inf 8900 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12890 df-struct 16480 df-ndx 16481 df-slot 16482 df-base 16484 df-plusg 16573 df-mulr 16574 df-sca 16576 df-vsca 16577 df-ip 16578 df-tset 16579 df-ple 16580 df-ds 16582 df-hom 16584 df-cco 16585 df-prds 16716 df-imas 16776 df-xps 16778 |
This theorem is referenced by: (None) |
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