Theorem evl1fval 19513

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		fvex 6113	. . . . . 6 ⊢ (Base‘𝑟) ∈ V
3	2	a1i 11	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) ∈ V)
4		id 22	. . . . . . . . 9 ⊢ (𝑏 = (Base‘𝑟) → 𝑏 = (Base‘𝑟))
5		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
6		evl1fval.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑅)
7	5, 6	syl6eqr 2662	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
8	4, 7	sylan9eqr 2666	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → 𝑏 = 𝐵)
9	8	oveq1d 6564	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑏 ↑𝑚 1𝑜) = (𝐵 ↑𝑚 1𝑜))
10	8, 9	oveq12d 6567	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑏 ↑𝑚 (𝑏 ↑𝑚 1𝑜)) = (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)))
11	8	mpteq1d 4666	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑦 ∈ 𝑏 ↦ (1𝑜 × {𝑦})) = (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))
12	11	coeq2d 5206	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1𝑜 × {𝑦}))) = (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦}))))
13	10, 12	mpteq12dv 4663	. . . . . 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})))))
14		simpl 472	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → 𝑟 = 𝑅)
15	14	oveq2d 6565	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (1𝑜 eval 𝑟) = (1𝑜 eval 𝑅))
16		evl1fval.q	. . . . . . 7 ⊢ 𝑄 = (1𝑜 eval 𝑅)
17	15, 16	syl6eqr 2662	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (1𝑜 eval 𝑟) = 𝑄)
18	13, 17	coeq12d 5208	. . . . 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))
19	3, 18	csbied 3526	. . . 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))
20		df-evl1 19502	. . . 4 ⊢ eval1 = (𝑟 ∈ V ↦ ⦋(Base‘𝑟) / 𝑏⦌((𝑥 ∈ (𝑏 ↑𝑚 (𝑏 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1𝑜 × {𝑦})))) ∘ (1𝑜 eval 𝑟)))
21		ovex 6577	. . . . . 6 ⊢ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ∈ V
22	21	mptex 6390	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∈ V
23		ovex 6577	. . . . . 6 ⊢ (1𝑜 eval 𝑅) ∈ V
24	16, 23	eqeltri 2684	. . . . 5 ⊢ 𝑄 ∈ V
25	22, 24	coex 7011	. . . 4 ⊢ ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∘ 𝑄) ∈ V
26	19, 20, 25	fvmpt 6191	. . 3 ⊢ (𝑅 ∈ V → (eval1‘𝑅) = ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∘ 𝑄))
27	1, 26	syl5eq 2656	. 2 ⊢ (𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∘ 𝑄))
28		fvprc 6097	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (eval1‘𝑅) = ∅)
29	1, 28	syl5eq 2656	. . . 4 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ∅)
30		co02 5566	. . . 4 ⊢ ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ∅) = ∅
31	29, 30	syl6eqr 2662	. . 3 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ∅))
32		df-evl 19328	. . . . . . 7 ⊢ eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟)))
33	32	reldmmpt2 6669	. . . . . 6 ⊢ Rel dom eval
34	33	ovprc2 6583	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (1𝑜 eval 𝑅) = ∅)
35	16, 34	syl5eq 2656	. . . 4 ⊢ (¬ 𝑅 ∈ V → 𝑄 = ∅)
36	35	coeq2d 5206	. . 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})))) ∘ ∅))
37	31, 36	eqtr4d 2647	. 2 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∘ 𝑄))
38	27, 37	pm2.61i 175	1 ⊢ 𝑂 = ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∘ 𝑄)

Syntax hints:  ¬ wn 3   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⦋csb 3499  ∅c0 3874  {csn 4125   ↦ cmpt 4643   × cxp 5036   ∘ ccom 5042  ‘cfv 5804  (class class class)co 6549  1_𝑜c1o 7440   ↑_𝑚 cmap 7744  Basecbs 15695   evalSub ces 19325   eval cevl 19326  eval₁ce1 19500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-evl 19328  df-evl1 19502

This theorem is referenced by:  evl1val  19514  evl1fval1lem  19515  evl1rhm  19517  pf1rcl  19534