nnf1o - Intuitionistic Logic Explorer

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Theorem nnf1o 11606

Description: Lemma for sum and product theorems. (Contributed by Jim Kingdon, 15-Aug-2022.)

**Hypotheses**
Ref	Expression
nnf1o.mn	⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
nnf1o.m	⊢ (𝜑 → 𝐹:(1...𝑀)–1-1-onto→𝐴)
nnf1o.n	⊢ (𝜑 → 𝐺:(1...𝑁)–1-1-onto→𝐴)

**Assertion**
Ref	Expression
nnf1o	⊢ (𝜑 → 𝑁 = 𝑀)

**Proof of Theorem nnf1o**
Step	Hyp	Ref	Expression
1		1zzd 9381	. . . 4 ⊢ (𝜑 → 1 ∈ ℤ)
2		nnf1o.mn	. . . . . 6 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
3	2	simprd 114	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
4	3	nnzd 9476	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ)
5	1, 4	fzfigd 10557	. . 3 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
6		nnf1o.m	. . . . 5 ⊢ (𝜑 → 𝐹:(1...𝑀)–1-1-onto→𝐴)
7		f1ocnv 5529	. . . . 5 ⊢ (𝐹:(1...𝑀)–1-1-onto→𝐴 → ^◡𝐹:𝐴–1-1-onto→(1...𝑀))
8	6, 7	syl 14	. . . 4 ⊢ (𝜑 → ^◡𝐹:𝐴–1-1-onto→(1...𝑀))
9		nnf1o.n	. . . 4 ⊢ (𝜑 → 𝐺:(1...𝑁)–1-1-onto→𝐴)
10		f1oco 5539	. . . 4 ⊢ ((^◡𝐹:𝐴–1-1-onto→(1...𝑀) ∧ 𝐺:(1...𝑁)–1-1-onto→𝐴) → (^◡𝐹 ∘ 𝐺):(1...𝑁)–1-1-onto→(1...𝑀))
11	8, 9, 10	syl2anc 411	. . 3 ⊢ (𝜑 → (^◡𝐹 ∘ 𝐺):(1...𝑁)–1-1-onto→(1...𝑀))
12	5, 11	fihasheqf1od 10915	. 2 ⊢ (𝜑 → (♯‘(1...𝑁)) = (♯‘(1...𝑀)))
13		nnnn0 9284	. . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
14		hashfz1 10909	. . 3 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(1...𝑁)) = 𝑁)
15	3, 13, 14	3syl 17	. 2 ⊢ (𝜑 → (♯‘(1...𝑁)) = 𝑁)
16	2	simpld 112	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ)
17		nnnn0 9284	. . 3 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
18		hashfz1 10909	. . 3 ⊢ (𝑀 ∈ ℕ₀ → (♯‘(1...𝑀)) = 𝑀)
19	16, 17, 18	3syl 17	. 2 ⊢ (𝜑 → (♯‘(1...𝑀)) = 𝑀)
20	12, 15, 19	3eqtr3d 2245	1 ⊢ (𝜑 → 𝑁 = 𝑀)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1372 ∈ wcel 2175 ^◡ccnv 4672 ∘ ccom 4677 –1-1-onto→wf1o 5267 ‘cfv 5268 (class class class)co 5934 1c1 7908 ℕcn 9018 ℕ₀cn0 9277 ...cfz 10112 ♯chash 10901

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-coll 4158 ax-sep 4161 ax-nul 4169 ax-pow 4217 ax-pr 4252 ax-un 4478 ax-setind 4583 ax-iinf 4634 ax-cnex 7998 ax-resscn 7999 ax-1cn 8000 ax-1re 8001 ax-icn 8002 ax-addcl 8003 ax-addrcl 8004 ax-mulcl 8005 ax-addcom 8007 ax-addass 8009 ax-distr 8011 ax-i2m1 8012 ax-0lt1 8013 ax-0id 8015 ax-rnegex 8016 ax-cnre 8018 ax-pre-ltirr 8019 ax-pre-ltwlin 8020 ax-pre-lttrn 8021 ax-pre-apti 8022 ax-pre-ltadd 8023

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-reu 2490 df-rab 2492 df-v 2773 df-sbc 2998 df-csb 3093 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-nul 3460 df-if 3571 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-int 3885 df-iun 3928 df-br 4044 df-opab 4105 df-mpt 4106 df-tr 4142 df-id 4338 df-iord 4411 df-on 4413 df-ilim 4414 df-suc 4416 df-iom 4637 df-xp 4679 df-rel 4680 df-cnv 4681 df-co 4682 df-dm 4683 df-rn 4684 df-res 4685 df-ima 4686 df-iota 5229 df-fun 5270 df-fn 5271 df-f 5272 df-f1 5273 df-fo 5274 df-f1o 5275 df-fv 5276 df-riota 5889 df-ov 5937 df-oprab 5938 df-mpo 5939 df-1st 6216 df-2nd 6217 df-recs 6381 df-frec 6467 df-1o 6492 df-er 6610 df-en 6818 df-dom 6819 df-fin 6820 df-pnf 8091 df-mnf 8092 df-xr 8093 df-ltxr 8094 df-le 8095 df-sub 8227 df-neg 8228 df-inn 9019 df-n0 9278 df-z 9355 df-uz 9631 df-fz 10113 df-ihash 10902

This theorem is referenced by: summodclem3 11610 prodmodclem3 11805

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