Theorem evl1fval 21838

Description: Value of the simple/same ring evaluation map. (Contributed by Mario Carneiro, 12-Jun-2015.)

Hypotheses

Ref	Expression
evl1fval.o	⊢ 𝑂 = (eval1‘𝑅)
evl1fval.q	⊢ 𝑄 = (1o eval 𝑅)
evl1fval.b	⊢ 𝐵 = (Base‘𝑅)

Assertion

Ref	Expression
evl1fval	⊢ 𝑂 = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄)

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝑄   𝑥,𝑅

Allowed substitution hints:   𝑄(𝑦)   𝑅(𝑦)   𝑂(𝑥,𝑦)

Proof of Theorem evl1fval

Dummy variables 𝑖 𝑟 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		evl1fval.o	. . 3 ⊢ 𝑂 = (eval1‘𝑅)
2		fvexd 6903	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) ∈ V)
3		id 22	. . . . . . . . 9 ⊢ (𝑏 = (Base‘𝑟) → 𝑏 = (Base‘𝑟))
4		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
5		evl1fval.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑅)
6	4, 5	eqtr4di 2790	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
7	3, 6	sylan9eqr 2794	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → 𝑏 = 𝐵)
8	7	oveq1d 7420	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑏 ↑m 1o) = (𝐵 ↑m 1o))
9	7, 8	oveq12d 7423	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑏 ↑m (𝑏 ↑m 1o)) = (𝐵 ↑m (𝐵 ↑m 1o)))
10	7	mpteq1d 5242	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})) = (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))
11	10	coeq2d 5860	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦}))) = (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))))
12	9, 11	mpteq12dv 5238	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑥 ∈ (𝑏 ↑m (𝑏 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})))) = (𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))))
13		simpl 483	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → 𝑟 = 𝑅)
14	13	oveq2d 7421	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (1o eval 𝑟) = (1o eval 𝑅))
15		evl1fval.q	. . . . . . 7 ⊢ 𝑄 = (1o eval 𝑅)
16	14, 15	eqtr4di 2790	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (1o eval 𝑟) = 𝑄)
17	12, 16	coeq12d 5862	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → ((𝑥 ∈ (𝑏 ↑m (𝑏 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})))) ∘ (1o eval 𝑟)) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄))
18	2, 17	csbied 3930	. . . 4 ⊢ (𝑟 = 𝑅 → ⦋(Base‘𝑟) / 𝑏⦌((𝑥 ∈ (𝑏 ↑m (𝑏 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})))) ∘ (1o eval 𝑟)) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄))
19		df-evl1 21826	. . . 4 ⊢ eval1 = (𝑟 ∈ V ↦ ⦋(Base‘𝑟) / 𝑏⦌((𝑥 ∈ (𝑏 ↑m (𝑏 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})))) ∘ (1o eval 𝑟)))
20		ovex 7438	. . . . . 6 ⊢ (𝐵 ↑m (𝐵 ↑m 1o)) ∈ V
21	20	mptex 7221	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∈ V
22	15	ovexi 7439	. . . . 5 ⊢ 𝑄 ∈ V
23	21, 22	coex 7917	. . . 4 ⊢ ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄) ∈ V
24	18, 19, 23	fvmpt 6995	. . 3 ⊢ (𝑅 ∈ V → (eval1‘𝑅) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄))
25	1, 24	eqtrid 2784	. 2 ⊢ (𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄))
26		fvprc 6880	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (eval1‘𝑅) = ∅)
27	1, 26	eqtrid 2784	. . . 4 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ∅)
28		co02 6256	. . . 4 ⊢ ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ ∅) = ∅
29	27, 28	eqtr4di 2790	. . 3 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ ∅))
30		df-evl 21627	. . . . . . 7 ⊢ eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟)))
31	30	reldmmpo 7539	. . . . . 6 ⊢ Rel dom eval
32	31	ovprc2 7445	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (1o eval 𝑅) = ∅)
33	15, 32	eqtrid 2784	. . . 4 ⊢ (¬ 𝑅 ∈ V → 𝑄 = ∅)
34	33	coeq2d 5860	. . 3 ⊢ (¬ 𝑅 ∈ V → ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ ∅))
35	29, 34	eqtr4d 2775	. 2 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄))
36	25, 35	pm2.61i 182	1 ⊢ 𝑂 = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄)

Syntax hints:  ¬ wn 3   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474  ⦋csb 3892  ∅c0 4321  {csn 4627   ↦ cmpt 5230   × cxp 5673   ∘ ccom 5679  ‘cfv 6540  (class class class)co 7405  1_oc1o 8455   ↑_m cmap 8816  Basecbs 17140   evalSub ces 21624   eval cevl 21625  eval₁ce1 21824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-evl 21627  df-evl1 21826

This theorem is referenced by:  evl1val  21839  evl1fval1lem  21840  evl1rhm  21842  pf1rcl  21859