nnf1o - Intuitionistic Logic Explorer

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Theorem nnf1o 11762

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 9419	. . . 4 ⊢ (𝜑 → 1 ∈ ℤ)
2		nnf1o.mn	. . . . . 6 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
3	2	simprd 114	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
4	3	nnzd 9514	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ)
5	1, 4	fzfigd 10598	. . 3 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
6		nnf1o.m	. . . . 5 ⊢ (𝜑 → 𝐹:(1...𝑀)–1-1-onto→𝐴)
7		f1ocnv 5547	. . . . 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 5557	. . . 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 10956	. 2 ⊢ (𝜑 → (♯‘(1...𝑁)) = (♯‘(1...𝑀)))
13		nnnn0 9322	. . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
14		hashfz1 10950	. . 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 9322	. . 3 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
18		hashfz1 10950	. . 3 ⊢ (𝑀 ∈ ℕ₀ → (♯‘(1...𝑀)) = 𝑀)
19	16, 17, 18	3syl 17	. 2 ⊢ (𝜑 → (♯‘(1...𝑀)) = 𝑀)
20	12, 15, 19	3eqtr3d 2247	1 ⊢ (𝜑 → 𝑁 = 𝑀)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1373 ∈ wcel 2177 ^◡ccnv 4682 ∘ ccom 4687 –1-1-onto→wf1o 5279 ‘cfv 5280 (class class class)co 5957 1c1 7946 ℕcn 9056 ℕ₀cn0 9315 ...cfz 10150 ♯chash 10942

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-coll 4167 ax-sep 4170 ax-nul 4178 ax-pow 4226 ax-pr 4261 ax-un 4488 ax-setind 4593 ax-iinf 4644 ax-cnex 8036 ax-resscn 8037 ax-1cn 8038 ax-1re 8039 ax-icn 8040 ax-addcl 8041 ax-addrcl 8042 ax-mulcl 8043 ax-addcom 8045 ax-addass 8047 ax-distr 8049 ax-i2m1 8050 ax-0lt1 8051 ax-0id 8053 ax-rnegex 8054 ax-cnre 8056 ax-pre-ltirr 8057 ax-pre-ltwlin 8058 ax-pre-lttrn 8059 ax-pre-apti 8060 ax-pre-ltadd 8061

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-reu 2492 df-rab 2494 df-v 2775 df-sbc 3003 df-csb 3098 df-dif 3172 df-un 3174 df-in 3176 df-ss 3183 df-nul 3465 df-if 3576 df-pw 3623 df-sn 3644 df-pr 3645 df-op 3647 df-uni 3857 df-int 3892 df-iun 3935 df-br 4052 df-opab 4114 df-mpt 4115 df-tr 4151 df-id 4348 df-iord 4421 df-on 4423 df-ilim 4424 df-suc 4426 df-iom 4647 df-xp 4689 df-rel 4690 df-cnv 4691 df-co 4692 df-dm 4693 df-rn 4694 df-res 4695 df-ima 4696 df-iota 5241 df-fun 5282 df-fn 5283 df-f 5284 df-f1 5285 df-fo 5286 df-f1o 5287 df-fv 5288 df-riota 5912 df-ov 5960 df-oprab 5961 df-mpo 5962 df-1st 6239 df-2nd 6240 df-recs 6404 df-frec 6490 df-1o 6515 df-er 6633 df-en 6841 df-dom 6842 df-fin 6843 df-pnf 8129 df-mnf 8130 df-xr 8131 df-ltxr 8132 df-le 8133 df-sub 8265 df-neg 8266 df-inn 9057 df-n0 9316 df-z 9393 df-uz 9669 df-fz 10151 df-ihash 10943

This theorem is referenced by: summodclem3 11766 prodmodclem3 11961

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