MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpsmnd0 Structured version   Visualization version   GIF version

Theorem xpsmnd0 18700
Description: The identity element of a binary product of monoids. (Contributed by AV, 25-Feb-2025.)
Hypothesis
Ref Expression
xpsmnd0.t 𝑇 = (𝑅 ×s 𝑆)
Assertion
Ref Expression
xpsmnd0 ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (0g𝑇) = ⟨(0g𝑅), (0g𝑆)⟩)

Proof of Theorem xpsmnd0
Dummy variables 𝑎 𝑏 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2724 . . 3 (Base‘𝑇) = (Base‘𝑇)
2 eqid 2724 . . 3 (0g𝑇) = (0g𝑇)
3 eqid 2724 . . 3 (+g𝑇) = (+g𝑇)
4 eqid 2724 . . . . . . 7 (Base‘𝑅) = (Base‘𝑅)
5 eqid 2724 . . . . . . 7 (0g𝑅) = (0g𝑅)
64, 5mndidcl 18674 . . . . . 6 (𝑅 ∈ Mnd → (0g𝑅) ∈ (Base‘𝑅))
76adantr 480 . . . . 5 ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (0g𝑅) ∈ (Base‘𝑅))
8 eqid 2724 . . . . . . 7 (Base‘𝑆) = (Base‘𝑆)
9 eqid 2724 . . . . . . 7 (0g𝑆) = (0g𝑆)
108, 9mndidcl 18674 . . . . . 6 (𝑆 ∈ Mnd → (0g𝑆) ∈ (Base‘𝑆))
1110adantl 481 . . . . 5 ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (0g𝑆) ∈ (Base‘𝑆))
127, 11opelxpd 5706 . . . 4 ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → ⟨(0g𝑅), (0g𝑆)⟩ ∈ ((Base‘𝑅) × (Base‘𝑆)))
13 xpsmnd0.t . . . . 5 𝑇 = (𝑅 ×s 𝑆)
14 simpl 482 . . . . 5 ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → 𝑅 ∈ Mnd)
15 simpr 484 . . . . 5 ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → 𝑆 ∈ Mnd)
1613, 4, 8, 14, 15xpsbas 17519 . . . 4 ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → ((Base‘𝑅) × (Base‘𝑆)) = (Base‘𝑇))
1712, 16eleqtrd 2827 . . 3 ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → ⟨(0g𝑅), (0g𝑆)⟩ ∈ (Base‘𝑇))
1816eleq2d 2811 . . . . 5 ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (𝑥 ∈ ((Base‘𝑅) × (Base‘𝑆)) ↔ 𝑥 ∈ (Base‘𝑇)))
19 elxp2 5691 . . . . . 6 (𝑥 ∈ ((Base‘𝑅) × (Base‘𝑆)) ↔ ∃𝑎 ∈ (Base‘𝑅)∃𝑏 ∈ (Base‘𝑆)𝑥 = ⟨𝑎, 𝑏⟩)
2014adantr 480 . . . . . . . . . 10 (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → 𝑅 ∈ Mnd)
2115adantr 480 . . . . . . . . . 10 (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → 𝑆 ∈ Mnd)
227adantr 480 . . . . . . . . . 10 (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (0g𝑅) ∈ (Base‘𝑅))
2311adantr 480 . . . . . . . . . 10 (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (0g𝑆) ∈ (Base‘𝑆))
24 simpl 482 . . . . . . . . . . 11 ((𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆)) → 𝑎 ∈ (Base‘𝑅))
2524adantl 481 . . . . . . . . . 10 (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → 𝑎 ∈ (Base‘𝑅))
26 simpr 484 . . . . . . . . . . 11 ((𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆)) → 𝑏 ∈ (Base‘𝑆))
2726adantl 481 . . . . . . . . . 10 (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → 𝑏 ∈ (Base‘𝑆))
28 eqid 2724 . . . . . . . . . . . 12 (+g𝑅) = (+g𝑅)
294, 28mndcl 18667 . . . . . . . . . . 11 ((𝑅 ∈ Mnd ∧ (0g𝑅) ∈ (Base‘𝑅) ∧ 𝑎 ∈ (Base‘𝑅)) → ((0g𝑅)(+g𝑅)𝑎) ∈ (Base‘𝑅))
3020, 22, 25, 29syl3anc 1368 . . . . . . . . . 10 (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → ((0g𝑅)(+g𝑅)𝑎) ∈ (Base‘𝑅))
31 eqid 2724 . . . . . . . . . . . 12 (+g𝑆) = (+g𝑆)
328, 31mndcl 18667 . . . . . . . . . . 11 ((𝑆 ∈ Mnd ∧ (0g𝑆) ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → ((0g𝑆)(+g𝑆)𝑏) ∈ (Base‘𝑆))
3321, 23, 27, 32syl3anc 1368 . . . . . . . . . 10 (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → ((0g𝑆)(+g𝑆)𝑏) ∈ (Base‘𝑆))
3413, 4, 8, 20, 21, 22, 23, 25, 27, 30, 33, 28, 31, 3xpsadd 17521 . . . . . . . . 9 (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (⟨(0g𝑅), (0g𝑆)⟩(+g𝑇)⟨𝑎, 𝑏⟩) = ⟨((0g𝑅)(+g𝑅)𝑎), ((0g𝑆)(+g𝑆)𝑏)⟩)
354, 28, 5mndlid 18679 . . . . . . . . . . 11 ((𝑅 ∈ Mnd ∧ 𝑎 ∈ (Base‘𝑅)) → ((0g𝑅)(+g𝑅)𝑎) = 𝑎)
3614, 24, 35syl2an 595 . . . . . . . . . 10 (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → ((0g𝑅)(+g𝑅)𝑎) = 𝑎)
378, 31, 9mndlid 18679 . . . . . . . . . . 11 ((𝑆 ∈ Mnd ∧ 𝑏 ∈ (Base‘𝑆)) → ((0g𝑆)(+g𝑆)𝑏) = 𝑏)
3815, 26, 37syl2an 595 . . . . . . . . . 10 (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → ((0g𝑆)(+g𝑆)𝑏) = 𝑏)
3936, 38opeq12d 4874 . . . . . . . . 9 (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → ⟨((0g𝑅)(+g𝑅)𝑎), ((0g𝑆)(+g𝑆)𝑏)⟩ = ⟨𝑎, 𝑏⟩)
4034, 39eqtrd 2764 . . . . . . . 8 (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (⟨(0g𝑅), (0g𝑆)⟩(+g𝑇)⟨𝑎, 𝑏⟩) = ⟨𝑎, 𝑏⟩)
41 oveq2 7410 . . . . . . . . 9 (𝑥 = ⟨𝑎, 𝑏⟩ → (⟨(0g𝑅), (0g𝑆)⟩(+g𝑇)𝑥) = (⟨(0g𝑅), (0g𝑆)⟩(+g𝑇)⟨𝑎, 𝑏⟩))
42 id 22 . . . . . . . . 9 (𝑥 = ⟨𝑎, 𝑏⟩ → 𝑥 = ⟨𝑎, 𝑏⟩)
4341, 42eqeq12d 2740 . . . . . . . 8 (𝑥 = ⟨𝑎, 𝑏⟩ → ((⟨(0g𝑅), (0g𝑆)⟩(+g𝑇)𝑥) = 𝑥 ↔ (⟨(0g𝑅), (0g𝑆)⟩(+g𝑇)⟨𝑎, 𝑏⟩) = ⟨𝑎, 𝑏⟩))
4440, 43syl5ibrcom 246 . . . . . . 7 (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (𝑥 = ⟨𝑎, 𝑏⟩ → (⟨(0g𝑅), (0g𝑆)⟩(+g𝑇)𝑥) = 𝑥))
4544rexlimdvva 3203 . . . . . 6 ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (∃𝑎 ∈ (Base‘𝑅)∃𝑏 ∈ (Base‘𝑆)𝑥 = ⟨𝑎, 𝑏⟩ → (⟨(0g𝑅), (0g𝑆)⟩(+g𝑇)𝑥) = 𝑥))
4619, 45biimtrid 241 . . . . 5 ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (𝑥 ∈ ((Base‘𝑅) × (Base‘𝑆)) → (⟨(0g𝑅), (0g𝑆)⟩(+g𝑇)𝑥) = 𝑥))
4718, 46sylbird 260 . . . 4 ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (𝑥 ∈ (Base‘𝑇) → (⟨(0g𝑅), (0g𝑆)⟩(+g𝑇)𝑥) = 𝑥))
4847imp 406 . . 3 (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ 𝑥 ∈ (Base‘𝑇)) → (⟨(0g𝑅), (0g𝑆)⟩(+g𝑇)𝑥) = 𝑥)
494, 28mndcl 18667 . . . . . . . . . . 11 ((𝑅 ∈ Mnd ∧ 𝑎 ∈ (Base‘𝑅) ∧ (0g𝑅) ∈ (Base‘𝑅)) → (𝑎(+g𝑅)(0g𝑅)) ∈ (Base‘𝑅))
5020, 25, 22, 49syl3anc 1368 . . . . . . . . . 10 (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (𝑎(+g𝑅)(0g𝑅)) ∈ (Base‘𝑅))
518, 31mndcl 18667 . . . . . . . . . . 11 ((𝑆 ∈ Mnd ∧ 𝑏 ∈ (Base‘𝑆) ∧ (0g𝑆) ∈ (Base‘𝑆)) → (𝑏(+g𝑆)(0g𝑆)) ∈ (Base‘𝑆))
5221, 27, 23, 51syl3anc 1368 . . . . . . . . . 10 (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (𝑏(+g𝑆)(0g𝑆)) ∈ (Base‘𝑆))
5313, 4, 8, 20, 21, 25, 27, 22, 23, 50, 52, 28, 31, 3xpsadd 17521 . . . . . . . . 9 (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (⟨𝑎, 𝑏⟩(+g𝑇)⟨(0g𝑅), (0g𝑆)⟩) = ⟨(𝑎(+g𝑅)(0g𝑅)), (𝑏(+g𝑆)(0g𝑆))⟩)
544, 28, 5mndrid 18680 . . . . . . . . . . 11 ((𝑅 ∈ Mnd ∧ 𝑎 ∈ (Base‘𝑅)) → (𝑎(+g𝑅)(0g𝑅)) = 𝑎)
5514, 24, 54syl2an 595 . . . . . . . . . 10 (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (𝑎(+g𝑅)(0g𝑅)) = 𝑎)
568, 31, 9mndrid 18680 . . . . . . . . . . 11 ((𝑆 ∈ Mnd ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑏(+g𝑆)(0g𝑆)) = 𝑏)
5715, 26, 56syl2an 595 . . . . . . . . . 10 (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (𝑏(+g𝑆)(0g𝑆)) = 𝑏)
5855, 57opeq12d 4874 . . . . . . . . 9 (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → ⟨(𝑎(+g𝑅)(0g𝑅)), (𝑏(+g𝑆)(0g𝑆))⟩ = ⟨𝑎, 𝑏⟩)
5953, 58eqtrd 2764 . . . . . . . 8 (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (⟨𝑎, 𝑏⟩(+g𝑇)⟨(0g𝑅), (0g𝑆)⟩) = ⟨𝑎, 𝑏⟩)
60 oveq1 7409 . . . . . . . . 9 (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥(+g𝑇)⟨(0g𝑅), (0g𝑆)⟩) = (⟨𝑎, 𝑏⟩(+g𝑇)⟨(0g𝑅), (0g𝑆)⟩))
6160, 42eqeq12d 2740 . . . . . . . 8 (𝑥 = ⟨𝑎, 𝑏⟩ → ((𝑥(+g𝑇)⟨(0g𝑅), (0g𝑆)⟩) = 𝑥 ↔ (⟨𝑎, 𝑏⟩(+g𝑇)⟨(0g𝑅), (0g𝑆)⟩) = ⟨𝑎, 𝑏⟩))
6259, 61syl5ibrcom 246 . . . . . . 7 (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥(+g𝑇)⟨(0g𝑅), (0g𝑆)⟩) = 𝑥))
6362rexlimdvva 3203 . . . . . 6 ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (∃𝑎 ∈ (Base‘𝑅)∃𝑏 ∈ (Base‘𝑆)𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥(+g𝑇)⟨(0g𝑅), (0g𝑆)⟩) = 𝑥))
6419, 63biimtrid 241 . . . . 5 ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (𝑥 ∈ ((Base‘𝑅) × (Base‘𝑆)) → (𝑥(+g𝑇)⟨(0g𝑅), (0g𝑆)⟩) = 𝑥))
6518, 64sylbird 260 . . . 4 ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (𝑥 ∈ (Base‘𝑇) → (𝑥(+g𝑇)⟨(0g𝑅), (0g𝑆)⟩) = 𝑥))
6665imp 406 . . 3 (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ 𝑥 ∈ (Base‘𝑇)) → (𝑥(+g𝑇)⟨(0g𝑅), (0g𝑆)⟩) = 𝑥)
671, 2, 3, 17, 48, 66ismgmid2 18593 . 2 ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → ⟨(0g𝑅), (0g𝑆)⟩ = (0g𝑇))
6867eqcomd 2730 1 ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (0g𝑇) = ⟨(0g𝑅), (0g𝑆)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wrex 3062  cop 4627   × cxp 5665  cfv 6534  (class class class)co 7402  Basecbs 17145  +gcplusg 17198  0gc0g 17386   ×s cxps 17453  Mndcmnd 18659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-tp 4626  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-er 8700  df-map 8819  df-ixp 8889  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-sup 9434  df-inf 9435  df-pnf 11248  df-mnf 11249  df-xr 11250  df-ltxr 11251  df-le 11252  df-sub 11444  df-neg 11445  df-nn 12211  df-2 12273  df-3 12274  df-4 12275  df-5 12276  df-6 12277  df-7 12278  df-8 12279  df-9 12280  df-n0 12471  df-z 12557  df-dec 12676  df-uz 12821  df-fz 13483  df-struct 17081  df-slot 17116  df-ndx 17128  df-base 17146  df-plusg 17211  df-mulr 17212  df-sca 17214  df-vsca 17215  df-ip 17216  df-tset 17217  df-ple 17218  df-ds 17220  df-hom 17222  df-cco 17223  df-0g 17388  df-prds 17394  df-imas 17455  df-xps 17457  df-mgm 18565  df-sgrp 18644  df-mnd 18660
This theorem is referenced by:  xpsinv  18980  rngqiprngimf1  21145
  Copyright terms: Public domain W3C validator