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Theorem rngccoALTV 42516
Description: Composition in the category of non-unital rings. (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
Hypotheses
Ref Expression
rngcbasALTV.c 𝐶 = (RngCatALTV‘𝑈)
rngcbasALTV.b 𝐵 = (Base‘𝐶)
rngcbasALTV.u (𝜑𝑈𝑉)
rngccofvalALTV.o · = (comp‘𝐶)
rngccoALTV.x (𝜑𝑋𝐵)
rngccoALTV.y (𝜑𝑌𝐵)
rngccoALTV.z (𝜑𝑍𝐵)
rngccoALTV.f (𝜑𝐹 ∈ (𝑋 RngHomo 𝑌))
rngccoALTV.g (𝜑𝐺 ∈ (𝑌 RngHomo 𝑍))
Assertion
Ref Expression
rngccoALTV (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺𝐹))

Proof of Theorem rngccoALTV
Dummy variables 𝑓 𝑔 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rngcbasALTV.c . . . 4 𝐶 = (RngCatALTV‘𝑈)
2 rngcbasALTV.b . . . 4 𝐵 = (Base‘𝐶)
3 rngcbasALTV.u . . . 4 (𝜑𝑈𝑉)
4 rngccofvalALTV.o . . . 4 · = (comp‘𝐶)
51, 2, 3, 4rngccofvalALTV 42515 . . 3 (𝜑· = (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st𝑣) RngHomo (2nd𝑣)) ↦ (𝑔𝑓))))
6 simprl 754 . . . . . . 7 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑣 = ⟨𝑋, 𝑌⟩)
76fveq2d 6336 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑣) = (2nd ‘⟨𝑋, 𝑌⟩))
8 rngccoALTV.x . . . . . . . 8 (𝜑𝑋𝐵)
9 rngccoALTV.y . . . . . . . 8 (𝜑𝑌𝐵)
10 op2ndg 7328 . . . . . . . 8 ((𝑋𝐵𝑌𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
118, 9, 10syl2anc 573 . . . . . . 7 (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
1211adantr 466 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
137, 12eqtrd 2805 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑣) = 𝑌)
14 simprr 756 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
1513, 14oveq12d 6811 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((2nd𝑣) RngHomo 𝑧) = (𝑌 RngHomo 𝑍))
166fveq2d 6336 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st𝑣) = (1st ‘⟨𝑋, 𝑌⟩))
17 op1stg 7327 . . . . . . . 8 ((𝑋𝐵𝑌𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
188, 9, 17syl2anc 573 . . . . . . 7 (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
1918adantr 466 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2016, 19eqtrd 2805 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st𝑣) = 𝑋)
2120, 13oveq12d 6811 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((1st𝑣) RngHomo (2nd𝑣)) = (𝑋 RngHomo 𝑌))
22 eqidd 2772 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔𝑓) = (𝑔𝑓))
2315, 21, 22mpt2eq123dv 6864 . . 3 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔 ∈ ((2nd𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st𝑣) RngHomo (2nd𝑣)) ↦ (𝑔𝑓)) = (𝑔 ∈ (𝑌 RngHomo 𝑍), 𝑓 ∈ (𝑋 RngHomo 𝑌) ↦ (𝑔𝑓)))
24 opelxpi 5288 . . . 4 ((𝑋𝐵𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
258, 9, 24syl2anc 573 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
26 rngccoALTV.z . . 3 (𝜑𝑍𝐵)
27 ovex 6823 . . . . 5 (𝑌 RngHomo 𝑍) ∈ V
28 ovex 6823 . . . . 5 (𝑋 RngHomo 𝑌) ∈ V
2927, 28mpt2ex 7397 . . . 4 (𝑔 ∈ (𝑌 RngHomo 𝑍), 𝑓 ∈ (𝑋 RngHomo 𝑌) ↦ (𝑔𝑓)) ∈ V
3029a1i 11 . . 3 (𝜑 → (𝑔 ∈ (𝑌 RngHomo 𝑍), 𝑓 ∈ (𝑋 RngHomo 𝑌) ↦ (𝑔𝑓)) ∈ V)
315, 23, 25, 26, 30ovmpt2d 6935 . 2 (𝜑 → (⟨𝑋, 𝑌· 𝑍) = (𝑔 ∈ (𝑌 RngHomo 𝑍), 𝑓 ∈ (𝑋 RngHomo 𝑌) ↦ (𝑔𝑓)))
32 simprl 754 . . 3 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑔 = 𝐺)
33 simprr 756 . . 3 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑓 = 𝐹)
3432, 33coeq12d 5425 . 2 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (𝑔𝑓) = (𝐺𝐹))
35 rngccoALTV.g . 2 (𝜑𝐺 ∈ (𝑌 RngHomo 𝑍))
36 rngccoALTV.f . 2 (𝜑𝐹 ∈ (𝑋 RngHomo 𝑌))
37 coexg 7264 . . 3 ((𝐺 ∈ (𝑌 RngHomo 𝑍) ∧ 𝐹 ∈ (𝑋 RngHomo 𝑌)) → (𝐺𝐹) ∈ V)
3835, 36, 37syl2anc 573 . 2 (𝜑 → (𝐺𝐹) ∈ V)
3931, 34, 35, 36, 38ovmpt2d 6935 1 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  cop 4322   × cxp 5247  ccom 5253  cfv 6031  (class class class)co 6793  cmpt2 6795  1st c1st 7313  2nd c2nd 7314  Basecbs 16064  compcco 16161   RngHomo crngh 42413  RngCatALTVcrngcALTV 42486
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-oadd 7717  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  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-hom 16174  df-cco 16175  df-rngcALTV 42488
This theorem is referenced by:  rngccatidALTV  42517  rngcsectALTV  42520  rhmsubcALTVlem4  42635
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