![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > xpsrngd | Structured version Visualization version GIF version |
Description: A product of two non-unital rings is a non-unital ring (xpsmnd 18705 analog). (Contributed by AV, 22-Feb-2025.) |
Ref | Expression |
---|---|
xpsrngd.y | ⊢ 𝑌 = (𝑆 ×s 𝑅) |
xpsrngd.s | ⊢ (𝜑 → 𝑆 ∈ Rng) |
xpsrngd.r | ⊢ (𝜑 → 𝑅 ∈ Rng) |
Ref | Expression |
---|---|
xpsrngd | ⊢ (𝜑 → 𝑌 ∈ Rng) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpsrngd.y | . . 3 ⊢ 𝑌 = (𝑆 ×s 𝑅) | |
2 | eqid 2731 | . . 3 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
3 | eqid 2731 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
4 | xpsrngd.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ Rng) | |
5 | xpsrngd.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Rng) | |
6 | eqid 2731 | . . 3 ⊢ (𝑥 ∈ (Base‘𝑆), 𝑦 ∈ (Base‘𝑅) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) = (𝑥 ∈ (Base‘𝑆), 𝑦 ∈ (Base‘𝑅) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) | |
7 | eqid 2731 | . . 3 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆) | |
8 | eqid 2731 | . . 3 ⊢ ((Scalar‘𝑆)Xs{〈∅, 𝑆〉, 〈1o, 𝑅〉}) = ((Scalar‘𝑆)Xs{〈∅, 𝑆〉, 〈1o, 𝑅〉}) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | xpsval 17523 | . 2 ⊢ (𝜑 → 𝑌 = (◡(𝑥 ∈ (Base‘𝑆), 𝑦 ∈ (Base‘𝑅) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) “s ((Scalar‘𝑆)Xs{〈∅, 𝑆〉, 〈1o, 𝑅〉}))) |
10 | 6 | xpsff1o2 17522 | . . . . 5 ⊢ (𝑥 ∈ (Base‘𝑆), 𝑦 ∈ (Base‘𝑅) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):((Base‘𝑆) × (Base‘𝑅))–1-1-onto→ran (𝑥 ∈ (Base‘𝑆), 𝑦 ∈ (Base‘𝑅) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) |
11 | 1, 2, 3, 4, 5, 6, 7, 8 | xpsrnbas 17524 | . . . . . 6 ⊢ (𝜑 → ran (𝑥 ∈ (Base‘𝑆), 𝑦 ∈ (Base‘𝑅) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) = (Base‘((Scalar‘𝑆)Xs{〈∅, 𝑆〉, 〈1o, 𝑅〉}))) |
12 | 11 | f1oeq3d 6830 | . . . . 5 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝑆), 𝑦 ∈ (Base‘𝑅) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):((Base‘𝑆) × (Base‘𝑅))–1-1-onto→ran (𝑥 ∈ (Base‘𝑆), 𝑦 ∈ (Base‘𝑅) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) ↔ (𝑥 ∈ (Base‘𝑆), 𝑦 ∈ (Base‘𝑅) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):((Base‘𝑆) × (Base‘𝑅))–1-1-onto→(Base‘((Scalar‘𝑆)Xs{〈∅, 𝑆〉, 〈1o, 𝑅〉})))) |
13 | 10, 12 | mpbii 232 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑆), 𝑦 ∈ (Base‘𝑅) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):((Base‘𝑆) × (Base‘𝑅))–1-1-onto→(Base‘((Scalar‘𝑆)Xs{〈∅, 𝑆〉, 〈1o, 𝑅〉}))) |
14 | f1ocnv 6845 | . . . 4 ⊢ ((𝑥 ∈ (Base‘𝑆), 𝑦 ∈ (Base‘𝑅) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):((Base‘𝑆) × (Base‘𝑅))–1-1-onto→(Base‘((Scalar‘𝑆)Xs{〈∅, 𝑆〉, 〈1o, 𝑅〉})) → ◡(𝑥 ∈ (Base‘𝑆), 𝑦 ∈ (Base‘𝑅) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):(Base‘((Scalar‘𝑆)Xs{〈∅, 𝑆〉, 〈1o, 𝑅〉}))–1-1-onto→((Base‘𝑆) × (Base‘𝑅))) | |
15 | f1of1 6832 | . . . 4 ⊢ (◡(𝑥 ∈ (Base‘𝑆), 𝑦 ∈ (Base‘𝑅) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):(Base‘((Scalar‘𝑆)Xs{〈∅, 𝑆〉, 〈1o, 𝑅〉}))–1-1-onto→((Base‘𝑆) × (Base‘𝑅)) → ◡(𝑥 ∈ (Base‘𝑆), 𝑦 ∈ (Base‘𝑅) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):(Base‘((Scalar‘𝑆)Xs{〈∅, 𝑆〉, 〈1o, 𝑅〉}))–1-1→((Base‘𝑆) × (Base‘𝑅))) | |
16 | 13, 14, 15 | 3syl 18 | . . 3 ⊢ (𝜑 → ◡(𝑥 ∈ (Base‘𝑆), 𝑦 ∈ (Base‘𝑅) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):(Base‘((Scalar‘𝑆)Xs{〈∅, 𝑆〉, 〈1o, 𝑅〉}))–1-1→((Base‘𝑆) × (Base‘𝑅))) |
17 | 2on 8486 | . . . . 5 ⊢ 2o ∈ On | |
18 | 17 | a1i 11 | . . . 4 ⊢ (𝜑 → 2o ∈ On) |
19 | fvexd 6906 | . . . 4 ⊢ (𝜑 → (Scalar‘𝑆) ∈ V) | |
20 | xpscf 17518 | . . . . 5 ⊢ ({〈∅, 𝑆〉, 〈1o, 𝑅〉}:2o⟶Rng ↔ (𝑆 ∈ Rng ∧ 𝑅 ∈ Rng)) | |
21 | 4, 5, 20 | sylanbrc 582 | . . . 4 ⊢ (𝜑 → {〈∅, 𝑆〉, 〈1o, 𝑅〉}:2o⟶Rng) |
22 | 8, 18, 19, 21 | prdsrngd 20077 | . . 3 ⊢ (𝜑 → ((Scalar‘𝑆)Xs{〈∅, 𝑆〉, 〈1o, 𝑅〉}) ∈ Rng) |
23 | eqid 2731 | . . . 4 ⊢ (◡(𝑥 ∈ (Base‘𝑆), 𝑦 ∈ (Base‘𝑅) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) “s ((Scalar‘𝑆)Xs{〈∅, 𝑆〉, 〈1o, 𝑅〉})) = (◡(𝑥 ∈ (Base‘𝑆), 𝑦 ∈ (Base‘𝑅) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) “s ((Scalar‘𝑆)Xs{〈∅, 𝑆〉, 〈1o, 𝑅〉})) | |
24 | eqid 2731 | . . . 4 ⊢ (Base‘((Scalar‘𝑆)Xs{〈∅, 𝑆〉, 〈1o, 𝑅〉})) = (Base‘((Scalar‘𝑆)Xs{〈∅, 𝑆〉, 〈1o, 𝑅〉})) | |
25 | 23, 24 | imasrngf1 20079 | . . 3 ⊢ ((◡(𝑥 ∈ (Base‘𝑆), 𝑦 ∈ (Base‘𝑅) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):(Base‘((Scalar‘𝑆)Xs{〈∅, 𝑆〉, 〈1o, 𝑅〉}))–1-1→((Base‘𝑆) × (Base‘𝑅)) ∧ ((Scalar‘𝑆)Xs{〈∅, 𝑆〉, 〈1o, 𝑅〉}) ∈ Rng) → (◡(𝑥 ∈ (Base‘𝑆), 𝑦 ∈ (Base‘𝑅) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) “s ((Scalar‘𝑆)Xs{〈∅, 𝑆〉, 〈1o, 𝑅〉})) ∈ Rng) |
26 | 16, 22, 25 | syl2anc 583 | . 2 ⊢ (𝜑 → (◡(𝑥 ∈ (Base‘𝑆), 𝑦 ∈ (Base‘𝑅) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) “s ((Scalar‘𝑆)Xs{〈∅, 𝑆〉, 〈1o, 𝑅〉})) ∈ Rng) |
27 | 9, 26 | eqeltrd 2832 | 1 ⊢ (𝜑 → 𝑌 ∈ Rng) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 Vcvv 3473 ∅c0 4322 {cpr 4630 〈cop 4634 × cxp 5674 ◡ccnv 5675 ran crn 5677 Oncon0 6364 ⟶wf 6539 –1-1→wf1 6540 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7412 ∈ cmpo 7414 1oc1o 8465 2oc2o 8466 Basecbs 17151 Scalarcsca 17207 Xscprds 17398 “s cimas 17457 ×s cxps 17459 Rngcrng 20053 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-2o 8473 df-er 8709 df-map 8828 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9443 df-inf 9444 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-fz 13492 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-hom 17228 df-cco 17229 df-0g 17394 df-prds 17400 df-imas 17461 df-xps 17463 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 df-minusg 18865 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 |
This theorem is referenced by: rngqiprng 21145 pzriprnglem1 21342 |
Copyright terms: Public domain | W3C validator |